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CORNELL
UNIVERSITY
LIBRARY
MATHEMATICS
Cornell University Library
QA 196.T13 1873
An elementary treatise on quaternions
3 1924 001 570 971
Moil
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QUATERNIONS
TAIT
SonDon : Cambridge warehouse,
17, PATERNOSTER ROW.
©amfiriDgc: deighton, bell, and co.
AN ELEMENTAEY TREATISE
ON
QUATEKNIONS
BY
P. G. TAIT, M.A.
FORMEKLY FELLOW OF SI. PETER'S COLLEGE, CAMBRIDGE
PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH
TfKpaKTVV,
nayav aevaov <f)V(rfas pif^iijiar txovaav.
SECOND EDITION, ENLARGED
AT THE TJNIVEESITY PRESS
[All Rights reserved,}
The original of tiiis book is in
tine Cornell University Library.
There are no known copyright restrictions in
the United States on the use of the text.
http://www.archive.org/details/cu31924001570971
PREFACE.
To THE first edition of this work, published in 1867, the following
was prefixed : —
' The present work was commenced in 1859, while I was a Pro-
fessor of Mathematics, and far more ready at Quaternion analysis
than I can now pretend to be. Had it been then completed I
should have had means of testing its teaching capabilities, and of
improving it, before publication, where found deficient in that
respect.
' The duties of another Chair, and Sir W. Hamilton's wish that
my volume should not appear till after the publication of his JSle-
ments, interrupted my already extensive preparations. I had worked
out nearly all the examples of Analytical Geometry in Todhunter's
Collection, and I had made various physical applications of the
Calculus, especially to Crystallography, to Geometrical Optics, and
to the Induction of Currents, in addition to those on Kinematics,
Electrodynamics, Fresnel's Wave Surface, &c., which are reprinted
in the present work from the Quarterly Mathematical Journal and
the Proceedings of the Royal Society of Edinburgh.
' Sir W. Hamilton, when I saw him but a few days before his
death, urged me to prepare my work as soon as possible, his being
almost ready for publication. He then expressed, more strongly
perhaps than he had ever done before, his profound conviction of
the importance of Quaternions to the progress of physical science ;
and his desire that a really elementary treatise on the subject should
soon be published.
VI PREFACE.
' I regret that I have so imperfectly fulfilled this last request of
my revered friend. When it was made I was already engaged,
along with Sir W. Thomson, in the laborious work of preparing
a large Treatise on Natural Philosophy. The present volume has
thus been written under very disadvantageous circumstances, espe-
cially as I have not found time to work up the mass of materials
which I had originally collected for it, but which I had not put
into a fit state for publication. I hope, however, that I have to
some extent succeeded in producing a thoroughly elementary work,
intelligible to any ordinary student; and that the numerous ex-
amples I have given, though not specially chosen so as to display
the full merits of Quaternions, will yet sufficiently shew their admir-
able simplicity and naturalness to induce the reader to attack the
Lectures and the Elements ; where he will find, in profusion, stores
of valuable results, and of elegant yet powerful analytical investiga-
tions, such as are contained in the writings of but a very few of the
greatest mathematicians. For a succinct account of the steps by
which Hamilton was led to the invention of Quaternions, and for
other interesting information regarding that remarkable genius, I
may refer to a slight sketch of his life and works in the North
British Review for September 1866.
' It will be found that I have not servilely followed even so great
a master, although dealing with a subject which is entirely his
own. I cannot, of course, tell in every case what I have gathered
from his published papers, or from his voluminous correspondence,
and what I may have made out for myself. Some theorems and
processes which I have given, though wholly my own, in the sense
of having been made out for myself before the publication of the
Elements, I have since found there. Others also may be, for 1 have
not yet read that tremendous volume completely, since much of it
bears on developments unconnected with Physics. But I have
endeavoured throughout to point out to the reader all the more
important parts of the work which I know to be wholly due to
Hamilton. A great part, indeed, may be said to be obvious to any
one who has mastered the preliminaries ; still I think that, in the
PREFACE. VU
two last Chapters especially, a good deal of original matter will be
found.
' The volume is essentially a working one, and, particularly in the
later Chapters, is rather a collection of examples than a detailed
treatise on a mathematical method. I have constantly aimed at
avoiding too great extension ; and in pursuance of this object have
omitted many valuable elementary portions of the subject. One of
these, the treatment of Quaternion logarithms and exponentials, I
greatly regret not having given. But if I had printed all that
seemed to me of use or interest to the student, I might easily have
rivalled the bulk of one of Hamilton's volumes. The beginner is
recommended merely to read the first five Chapters, then to work
at Chapters VI, VII, VIII (to which numerous easy Examples are
appended). After this he may work at the first five, with their
(more difficult) Examples ; and the remainder of the book should
then present no difficulty.
' Keeping always in view, as the great end of every mathematical
method, the physical applications, I have endeavoured to treat the
subject as much as possible from a geometrical instead of an analy-
tical point of view. Of course, if we premise the properties of i,j, k
merely, it is possible to construct from them the whole system* ;
just as we deal with the imaginary of Algebra, or, to take a closer
analogy, just as Hamilton himself dealt with Couples, Triads, and
Sets. This may be interesting to the pure analyst, but it is repulsive
to the physical student, who should be led to look upon i, _/, k from
the very first as geometric realities, not as algebraic imaginaries.
' The most striking peculiarity of the Calculus is that muUipli-
cation is not generally commutative, i.e. that qr is in general different
from rq, r and q being quaternions. Still it is to be remarked that
something similar is true, in the ordinary coordinate methods, of
operators and functions : and therefore the student is not wholly
unprepared to meet it. No one is puzzled by the fact that log.cos.a;
* This has been done by Hamilton himself, as one among many methods he has
employed ; and it is also the foundation of a memoir by M. AU^gret, entitled Esmi
sv/r le Calcul des Quaternions (Paris, 1862).
viii PREFACE.
is not equal to cos.log.a?, or that v/ j- is not equal to -^s/V'
Sometimes, indeed, this rule is most absurdly violated, for it is
usual to taJie cos^a; as equal to (cos xY, while cos-^a; is not equal to
(cos «)"■'. No such incongruities appear in Quaternions j but what
is true of operators and functions in other methods, that they are
not generally commutative, is in Quaternions true in the multipli-
cation of (vector) coordinates.
' It will be observed by those who are acquainted with the Cal-
culus that I have, in many cases, not given the shortest or simplest
proof of an important proposition. This has been done with the
view of including, in moderate compass, as great a variety of
methods as possible. With the same object I have endeavoured to
supply, by means of the Examples appended to each Chapter, hints
(which will not be lost to the intelligent student) of farther develop-
ments of the Calculus. Many of these are due to Hamilton, who,
in spite of his great originality, was one of the most excellent
examiners any University can boast of.
' It must always be remembered that Cartesian methods are mere
particular cases of Quaternions, where most of the distinctive fea-
tures have disappeared; and that when, in the treatment of any
particular question, scalars have to be adopted, the Quaternion
solution becomes identical with the Cartesian one.. Nothing there-
fore is ever lost, though much is generally gained, by employing
Quaternions in preference to ordinary methods. In fact, even when
Quaternions degrade to scalars, they give the solution of the most
general statement of the problem they are applied to, quite inde-
pendent of any limitations as to choice of particular coordinate
axes.
'There is one very desirable object which such a work as this
may possibly fulfil. The University of Cambridge, while seeking
to supply a real want (the deficiency of subjects of examination for
mathematical honours, and the consequent frequent introduction of
the wildest extravagance in the shape of data for " Problems "), is
in danger of making too much of such elegant trifles as Trilinear
PREFACE. IX
Coordinates, while gigantic systems like Invariants (which, by the
way, are as easily introduced into Quaternions as into Cartesian
methods) are quite beyond the amount of mathematics which even
the best students can master in three years' reading. One grand
step to the supply of this want is, of course, the introduction into
the scheme of examination of such branches of mathematical physics
as the Theories of Heat and Electricity. But it appears to me that
the study of a mathematical method like Quaternions, which, while
of immense power and comprehensiveness, is of extraordinary sim-
plicity, and yet requires constant thought in its applications, would
also be of great benefit. With it there can be no " shut your eyes,
and write down your equations," for mere mechanical dexterity of
analysis is certain to lead at once to error on account of the novelty
of the processes employed.
'The Table of Contents has been drawn up so as to give the
student a short and simple summary of the chief fundamental for-
mulae of the Calculus itself, and is therefore confined to an analysis
of the first five [and the two last] chapters.
' In conclusion, I have only to say that I shall be much obliged
to any one, student or teacher, who will point out portions of the
work where a difficulty has been found ; along with any inaccuracies
which may be detected.. As I have had no assistance in the revision
of the proof-sheets, and have composed the work at irregular in-
tervals, and while otherwise laboriously occupied, I fear it may
contain many slips and even errors. Should it reach another edition
there is no doubt that it will be improved in many important par-
ticulars.'
To this I have now to add that I have been equally surprised
and delighted by so speedy a demand for a second edition — and the
more especially as I have had many pleasing proofs that the
work has had considerable circulation in America. There seems
now at last to be a reasonable hope that Hamilton's grand in-
vention will soon find its way into the working world of science,
to which it is certain to render enormous services, and not be laid
X PREFACE.
aside to be unearthed some centuries hence by some grubbing
antiquary.
It can hardly be expected that one whose time is mainly en-
grossed by physical science, should devote much attention to the
purely analytical and geometrical applications of a subject like this ;
and I am conscious that in many parts of the earlier chapters I
have not fully exhibited the simplicity of Quaternions. I hope,
however, that the corrections and extensions now made, especially
in the later chapters, will render the work more useful for my chief
object, the Physical Applications of Quaternions, than it could have
been in its first crude form.
I have to thank various correspondents, some anonymous, for
suggestions as well as for the detection of misprints and slips of
the pen. The only absolute error which has been pointed out to
me is a comparatively slight one which had escaped my own notice :
a very grave blunder, which I have now corrected, seems not to
have been detected by any of my correspondents, so that I cannot
be quite confident that others may not exist.
I regret that I have not been able to spare time enough to re-
write the work ; and that, in consequence of this, and of the large
additions which have been made (especially to the later chapters),
the whole will now present even a more miscellaneously jumbled
appearance than at first.
It is well to remember, however, that it is quite possible to
make a book too easy reading, in the sense that the student may
read it through several times without feeling those difficulties
which (except perhaps in the case of some rare genius) must
attend the acquisition of really useful knowledge. It is better to
have a rough climb (even cutting one's own steps here and there)
than to ascend the dreary monotony of a marble staircase or a
well-made ladder. Royal roads to knowledge reach only the par-
ticular locality aimed at — and there are no views by the way.
It is not on them that pioneers are trained for the exploration of
unknown regions.
But I am happy to say that the "possible repulsiveness of my
PREFACE. xi
early chapters cannot long- be advanced as a reason for not at-
tacking this fascinating subject. A still more elementary work
than the present will soon appear, mainly from the pen of my
colleague Professor Kelland. In it I give an investigation of
the properties of the linear and vector function, based directly
upon the Kineinatics of Homogeneous Strain, and therefore so
different in method from that employed in this work that it may
prove of interest to even the advanced student.
Since the appearance of the first edition I have managed (at least
partially) to effect the application of Quaternions to line, surface,
and volume integrals, such as occur in Hydrokinetics, Electricity,
and Potentials generally. I was first attracted to the study of
Quaternions by their promise of usefulness in such applications,
and, though I have not yet advanced far in this new track, I have
got far enough to see that it is certain in time to be of incalculable
value to physical science. I have given towards the end of the
work all that is necessary to put the student on this track, which
will, I hope, soon be followed to some purpose.
One remark more is necessary. I have employed, as the positive
direction of rotation, that of the earth about its axis, or about the
sun, as seen in our northern latitudes, i.e. that opposite to the direc-
tion of motion of the hands of a watch. In Sir W. Hamilton's
great works the opposite is employed. The student will find no
difficulty in passing from the one to the other ; but, without pre-
vious warning, he is liable to be much perplexed.
With regard to notation, I have retained as nearly as possible
that of Hamilton, and where new notation was necessary I have
tried to make it as simple and as little incongruous with Hamil-
ton's as possible. This is a part of the work in which great care
is absolutely necessary; for, as the subject gains development,
fresh notation is inevitably required ; and our object must be to
make each step such as to defer as long as possible the revolution
which must ultimately come.
Many abbreviations are possible, and sometimes very useful in
private work ; but, as a rule, they are un suited for print. Every
xii PREFACE.
analyst, like every short-hand writer, has his own special con-
tractions ; but, when he comes to publish his results, he ought
invariably to put such devices aside. If all did not use a com-
mon mode of public expression, but each were to print as he is
in the habit of writing for his own use, the confusion would be
utterly intolerable.
Finally, I must express my great obligations to my friend
M. M. U. Wilkinson of Trinity College, Cambridge, for the care
with which he has read my proofs, and for many valuable sug-
gestions.
P. G. TAIT.
College, EDnfBUROH,
Octoher 1873.
CONTENTS.
Chapter I. — Vectoks and their Composition 1-22
Sketch of the attempts made to represent geometrically the unaginary of
algebra. §§ 1-13.
De Moivre's Theorem interpreted in plane rotation. § 8.
Curious speculation of Servois. §11.
Elementary geometrical ideas connected 'with relative position. § 15.
Definition of a Vbctoe. It may be employed to denote translation. § 16.
Expression of a vector by one symbol, containing implicitly three distinct
numbers. Extension of the signification of the symbol = . § IS.
The sign + defined in accordance with the interpretation of a rector as
representing translation. § 19.
Definition of - . It simply reverses a vector. § 20.
Triangles and polygons of vectors, analogous to those of forces and of simul-
taneous velocities. § 31.
When two vectors are paralkl we have
a = xp. § 22.
Any vector whatever may be expressed in terms of three distinct vectors,
which are not coplana, by the formula
p = xa+yP+zy,
which exhibits the three numbers on which the vector depends. § 23.
Any vector in the same plane with a and /S may be written
p = xa+yp. §24.
The equation 'sr = p,
between two vectors, is equivalent to three distinct equations among
numbers. § 25.
The Oonmmtative and Associative Laws hold in the combination of vectors by
the signs + and - . § 27.
The equation p = »/S,
where ^ is a variable, and p a fixed, vector, represents a line drawn
through the origin parallel to j3.
p = O + K/S
is the equation of a line drawn through the extremity of a and parallel
to jS, § 28.
p = ya+x§
represents the plane through the origin parallel to a and p. § 29.
xiv CONTENTS.
The condition that p, a, /3 may terminate in the same line is
p/) + jo + r/S = 0,
subject to the identical relation
Similarly pp + qa + r§ + ay = 0,
with p + q^ + r-\rs = 0,
is the condition that the extremities of four vectors lie in one plane. § 30.
Examples with solutions. § 81.
Differentiation of a vector, when given as a function of one number. §§ 32-38.
If the equation of a curve be
p = 4>{s)
where s is the length of the arc, dp is a vector tangent to the curve, and
its length is ds. §§ 38, 39.
Examples with solutions. §§ 40-44.
Examples to Chapter 1 22-24
Chapter II. — Products and Quotients of Vectors . . . 25-46
Here we begin to see what a quaternion is. When two vectors are parallel
their quotient is a number. §§ 45, 46.
When they are perpendicular to one another, their quotient is a vector per-
pendicular to their plane. § 47, 72.
When they are neither parallel nor perpendicular the quotient in general
involves fovtr distinct numbers — and is thus a Quatbbnion. § 47.
A quaternion regarded as the operator which turns one vector into another.
It is thus decomposable into two factors, whose order is indifferent, the
stretching factor or Tenbob, and the turning factor or Versob. These
are denoted by Tq, and Uq. § 48.
The equation /3 = j a
o
gives = It or /3a~' = g, hiit not in general
nr^^ = q. §49.
q or j3a~' depends only on the relative lengths, and directions, of jS and a.
§60.
Reci^ocal of a quaternion defined,
2=-g,ves-orgi = -,
y-2-' = -^. U.q-^ = {Uq)-\ §51.
Definition of the Conjugate of a quaternion,
Kq={Tqyqr\
and qKq = Kq.q = [Tqf. § 52.
Eepresentation of versors by arcs on the unit-sphere. § 53.
Versor multiplication illustrated by the composition of arcs. § 54.
Proof that K{qr) = Kr . Kq. § 55.
Proof of the Associative Law of Multiplication
p.qr=^pq.r. §§57-60.
[Digression on Spherical Conies. § 59'.]
CONTENTS. XV
Quaternion addition and subtraction are commutative. § 61.
Quaternion multiplication and division are disti-^uiive. § 62.
CompoEdtion of quadrantal veraors in planes at right angles to each other.
Calling them i, j, k, we have
i'=f=k^= - 1, »)■= -ji = h, jh= -hj=i, K= -ilc=j,
yh=-l. §§64-71.
A unit-vector, when employed as a factor, may be considered as a quadrautal
versor whose plane is perpendicular to the vector, Hence the equations
just written are true of any set of rectangular unit-vectors i, j, Js. § 72.
The product, and the quotient, of two vectors at right angles to each other is
a third perpendicular to both. Hence
Ka = -a,
and {Ta)^ = aKa=-a'. §73.
Every versor may be expressed as a power of some unit-vector. § 74.
Every quaternion may be expressed as a power of a vector. § 73.
The Index Law is true of quaternion multiplication and division. § 76.
Quaternion considered as the sum of a SOALAB and Yeoiob.
q = ^ = x+y = Si + Vi. §77.,
a
Proof that SKq = Sq, YKq = -7q, §79.
Quadrinomial expression for a quaternion
q = w+ix+jy + Jcz.
An equation between quaternions is equivalent to four equations between
numbers (or scalars). § 80.
Second proof of the distributive law of multiplication. § 81.
Algebraic determination of the constituents of the product and quotient of two
vectors. §§ 82-84.
Second proof of the associative law of multiplication. § 85.
Proof of the formulae SajS = S^a,
FajS = - rpa,
o/S = K^a,
S.qrs = S.rsq = S.sqr,
S. a/Sv = S.pya = iS'.70jS = - S^ ayp = &c. §§ 86-89.
Proof of the formulae
V.aVpy = ySaP-pSya,
V. 0JS7 = aSpy - pSya + 7/S0/S;
7.0/87= ^-Y/So,
V. FajS Vyd = o<S'.;875 - 185.07S,
= SS.aPy-yS.a^S,
SS.aPy = aS.pyS + pS.yaS + yS.apS,
= VapSyS+ r§ySad+ VyaSpS. §§ 90-92.
Hamilton's proof that the product of two parallel vectors must be a scalar, and
that of perpendicular vectors, a vector; if quaternions are to deal with
space indifferently in all directions. § 93.
Examples to Chaptek II 46, 47
xvi CONTENTS.
Chapter III. — Interpretations and Transformations of
Quaternion Expressions 48-67
If 6 be the angle between two vectors, a and j9, we have
S^ = ^cos e, SaB = - TaT^ cosff,
o xo
a Ta
Applications to plane trigonometry. §§ 94-97.
shews that o is perpendicular to jS, while
Fo/3 = 0,
shews that a and p are parallel.
S.aPy
is the volume of the parallelepiped three of whose conterminous edges are
a, jS, 7. Hence S.aPy =
shews that a, j3, 7 are coplanar.
Expression of S. apy as a determinant. §§ 98-102.
Proof that {Tg)' = {Si)'+ {TVq)",
and T{qr) = Tq, Tr. % 103,
Simple propositions in plane trigonometry. § 104.
Proof that - apa~^ ia the vector reflected ray, when j3 is the incident ray and o
normal to the reflecting surface. § 105.
Interpretation of 0/87 when it is a vector. § 106.
Examples of variety in simple transformations. § 107.
Introduction to spherical trigonometry. §§ 108-113.
Bepreaentation, graphic, and by quaternions, of the spherical excess. §g 114, IIS.
Loci represented by different equations — points, lines, surfaces, and solids.
§§ 116-119.
Proof that r-i (rV)* g-i = U(rq + KrKq). § 120.
Proof of the transformation
(Sv^pf + (S^pf + (Sypf = i^^^y,
-''^ M«-}=^"(v55l>«-v^>7). ^121.122.
BlQUATEENlONS. §§ 123-125.
Convenient abbreviations of notation. §§ 126, 127.
Examples to Chaptbe III 68-70
Chapter IV. — Differentiation op Quaternions .... 71-76
Definition of a dififerential,
where dq is any quaternion whatever.
We may write dFq =f{q, dq),
where / is linear and homogeneous in dq; but we cannot generally write
dFq = f{q)dq. §§128-131.
CONTENTS. xvii
Definition of the differential of aTTunction of more qnatemions ttan one.
d(qr) = qdr + dq.r, but not generally d($r) = qdr + rdq. § 132.
Proof that ^ = S^,
Tp p
^=F^,&c. §133.
Up p
Successive differentiation; Taylor's theorem. §§ 134, 135.
If the equation of a surface be
-P0>) = c,
the differential may be written
Svdp = 0,
where >< is a vector normal to the surface. § 137.
Examples to Chaptbe IV 76
Chaptee V. — The Solution op Equations op the First Degree.
77-100
The most general equation of the first degree in an unknown quaternion q,
may be written 2 V. aqb + S .cq = d,
where a, h, c, d are given quaternions. Elimination of ;S'}, and reduction
to the vector equation
<1>P = S. aSPp = y. |§ 138, 139.' : -^
General proof that ^'p is expressible as a linear function of p, <pp, and <p'p.
§liO.
Value of <l> for an ellipsoid, employed to illustrate the general theory.
§§ 141-143.
Hamilton's solution of (pp = y.
If we write Sa<j>p = Sp(p'a,
the functions <p and ^' are said to be conjugate, and
m^-^V\ii = T<t>'\<t>'p..
Proof that m, whose value may be written as
S .<p'K<p! fup'v
is the same for all values of \, n,v. §§ 144-146.
Proof that if »n^ = m + jHi jr + 7»j jr» + j',
S (\<p'iup'v + f'Xfup'v + <p'\<p'ia/)
where m^ =
and
S.XiJiv
8 (X^0V + <l>'\iiv + K(p'nv)
' S.\iiv
(which, like m, are Invarianti,)
then mg (<p + g)~^ VXn = (m^-' -k-gx + f) ^^f-
Also that X = '»a— ■/>>
whence the final form of solution
m<p-^ = m.i-mi<p + <p''. §§147,148.
Examples. § 149-161.
b
xviii CONTENTS.
The fundamental cubic
(/I'-mjif' + m.^-m = (<f-£rj) (^-ffi)("?'-ff.) = 0.
When is its own conjugate, the roots of the cubic are real ; and the
equation ^Plip. = 0>
or (.<p-g) P = 0,
is satisfied by a set of three real and mutually perpendicular vectors.
Geometrical interpretation of these results. §§ 162-166.
Proof of the transformation
i>p =fp + hV. (i + ek)'f (i—ek)
where (<^— ffi)* = 0,
C — )
Another transformation is
(pp = aaVap + bPSPp. §§167-169.
Other properties of i^. Proof that
Sp(<t> + g)~*P = 0) and Sp (<p + h)~^p =
represent the same surface if
mSp(p~^p = ghp^.
Proof that when ip is not self-conjugate
ipp = (p'p + Vep.
Proof that, if q = a(pa + 0(p0 + 71^7,
where a, P, 7 are any rectangular unit-vectors whatever, we have
Sq— — TOj, Vq = f.
This quaternion can be expressed in the important form
2 = v#. §§ 170-174.
Degrees of indeterminateness of the solution of a quaternion equation —
Examples. §§ 176-179.
The linear function of a quaternion is given by a symbolical biquadratic.
§180.
Particular forms of linear equations. §§ 181-183.
A quaternion equation of the mth degree in general involves a scalar equation
of degree m*. § 184.
Solution of the equation ^ = qa + T>. §185.
Examples to ChaptebV 101-103
Chapter VI. — Gbometei of the Straight Line and Plane ,
104-117
Examples to Chafteb VI 117-119
Chapter VII. — ^The Sphere and Cyclic Cone . . . 120-132
EXAMPLES TO Chapteb VII 132-134
Chapter VIII. — Surfaces of the Second Order . . 135-151
Examples xo Chapter VIII 151-154
CONTENTS. XIX
Chapter IX. — Geometry op (?urves and Surfaces . 155-186
Examples to Chapter IX 187-194
Chapter X. — Kinematics 195-218
If p = 0< be the vector of a moving point in terms of the time, p is the
vector velocity, and p the vector acceleration.
(T = p = ((>'(<) is the equation of the Hodograph.
p = vp' + v'p" gives the normal and tangential accelerations.
Vpp = if acceleration directed to a point, whence Tpp = y.
Examples. — Planetary acceleration. Here the equation is
/.Dp
given Vp^ — y ; whence the hodograph is
p = ty~'^—iiUp.y~\
and the orbit is the section of
j«r/. = Sf(7='£->-p)
by the plane Syp = 0.
Epitrochoids, &c. §| 336-348.
Rotation of a rigid system. Composition of rotations. The operator 5s( )q—^
turns the system it is applied to through 2 b times the angle of g, about
the axis of q. If the position of a system at time t is derived from the
initial position by j ( ) 2~', the instantaneous axis is
€ = 2Vqg-^. §§ 349-359.
Homogeneous strain. Criterion of pure strain. Separation of the rotational
jrom the pure part. Extraction of the square root of a strain. A strain
^ is equivalent to a, pure strain V*^'^ followed by the rotation — - — .
Simple Shear. §§ 360-367. '^'P''!'
Displacements of systems of points. Consequent condensation and rotation.
Preliminary about the use of V. §§ 368-371.
Moment of inertia. § 372.
Examples to Chapter X 218-221
Chapter XI. — Physical Applications 222-288
Condition of equilibrium of -a rigid system is 'SS.PSa = 0, where j8 is a vector
force, a its point of application. Hence the usual six equations in the
form 2j8 = 0, SVa0 = 0. Central axis, &c. §§ 373-378.
For the motion of a rigid system
SS(md-P)Sa = 0,
whence the usual forms. The equation
2j= q<p-^{q-^yq),
where y is given in terms of t and q if forces act, but is otherwise constant,
contains the whole theory of the motion of a rigid body with one point
fixed. Reduction to the ordinary form
dt dm dx _dy _ dz
"2 W~X~T~ ~Z'
Here, if no forces act, W, X, T, Z are homogeneous functions of the third
degree in w, x, y, z. Equation for precession. §§ 379-401.
General equation of motion of simple pendulum. Foucault's pendulum.
§§ 402-406.
b3
CONTENTS.
Problem on reflecting surfaces. § 406.
Freanel's Theory of Double Eefraction. Various fonns of the equation of
Fresnel's Wave-surface ;
S.p(.f-p»)-V = -l. T(p-'-<l>-')-ip = 0, l:=-pp' + (,T±S)VKpViip,
The conical cusps and circles of contact. Lines of vibration, &c. §§
407-427.
Electrodynamics. The vector action of a closed circuit on an element of
current o, is proportional to Vai0 where
^ rVada CdUa
the integration extending round the circuit. Mutual action of two closed
circuits, and of solenoids. Mutual action of magnets. Potential of a
closed circuit. Magnetic curves. §§ 428-448.
Physical applications of
, d . d , i
dx dy dz
Effect of V on various functions of p. = kc +jy + kz.
2
V/)=-3, VTp=V'p, vTJp = —~. V8ap=-a, v7ap=2a.
Applications of the theorem
S.SpV^ = SS.oV^ . §§ 449-457.
Jp» Ip
Farther examination of the use of V as applied to displacements of groups of
points. Proof of the fundamental theorem for comparing an integral
over a closed surface with one through its content
///S.V<rd^=//S.aUvd8.
Hence Green's Theorem. Limitations and ambiguities. §§ 458-476.
Similar theorem for double and single integrals
fS.adp =//S.UV7ads.
Applications of these to distributions of magnetism, and to Ampere's
Directrice. Also to the Stress-function. §5 477-491.
e-S<rV/(p)= f(p + <r).
Applications and consequences. Separation of symbols of operation, and
their treatment as quantities. §§ 492-495.
Applications of V in connection with the Calculus of Variations. If
A =/QTdp, SA=0 gives ^(,Qp')-vQ = 0.
Ui8
Applications to Varying Action, Brachistochrones, Catenaries. §§ 496-504.
Thomson's Theorem that there is one and but one solution of
S.VCe'Vit) = 4irr. §505.
MiSCELLANEOtrS EXAMPLES 288-296
ERRATUM.
Page 102, line 20, for ^p—tpipp read <j>if/p—</«l>'p.
QUATERNIONS.
CHAPTER I.
VECTORS, AND THEIR COMPOSITION.
1,] For more than a century and a half the geometrical re-
presentation of the negative and imaginary algebraic quantities,
— 1 and a/— 1, or, as some prefer to write them, — and — *, has
been a favourite subject of speculation with mathematicians. The
essence of almost all of the proposed processes consists in em-
ploying such expressions to indicate the direction, not the length,
of lines.
2.] Thus it was long ago seen that if positive quantities were
measured o£F in one direction along a fixed line, a useful and lawful
convention enabled us to express negative quantities of the same
kind by simply laying them off on the same line in the opposite
direction. This convention is an essential part of the Cartesian
method, and is constantly employed in Analytical Geometry and
Applied Mathematics.
3.] WaUis, towards the end of the seventeenth century^ proposed
to represent the impossible roots of a quadratic equation by going
out of the line on which, if real, they would have been laid off.
His construction is equivalent to the consideration of v — 1 as a
directed unit-line perpendicular to that on which real quantities
are measured.
4. J In the usual notation of Analytical ^Geometry of two
dimensions, when rectangular axes are employed, this amounts
to reckoning each unit of length along Oy as +v— 1, and on
Oy' as — V — 1 ; whUe on Ox each unit is +1, and on Oaf it is
B
2 QUATEKNIONS. [5.
— 1 . If we look at these four lines in circular order, i. e. in the
order of positive rotation (opposite to that of the hands of a watch),
they give r—r _ _ y3"i
In this series each expression is derived from that which precedes
it by multiplication by the factor v— 1. Hence we may consider
-v/— 1 as an operator, analogous to a handle perpendicular to the
plane of ay, whose effect on any line is to make it rotate (positively)
about the origin through an angle of 90°.
5.] In such a system^ a point is defined by a single imaginary
expression. Thus a + b v — 1 may be considered as a single quan-
tity, denoting the point whose coordinates are a and b. Or, it may
be used as an expression for the line joining that point with the
origin. In the latter sense, the expression a + b \/—l implicitly
contains the direction, as well as the length, of this line ; since, as
we see at once, the direction is inclined at- an angle tan^^- to the
axis oi X, and the length is \/a^ + J^.
6.] Operating on this symbol by the factor V— 1, it becomes
— 3-|-a\/— 1 ; and now, of course, denotes the point whose x avAy
coordinates are —b and a ; or the line joining this point with the
origin. The length is still Va^ + b"^, but the angle the line makes
with the axis of a; is tan~^ (~ 7") ' 'w^^'ich is evidently 90° greater
than before the operation.
7.3 De Moivre's Theorem tends to lead us still farther in the
same direction. In fact, it is easy to see that if we use, instead
of >/— 1, the more general factor cosa+ ^/— 1 sin a, its effect on
any line is to turn it through the (positive) angle a in the plane
oix,y. [Of course the former factor, 'J —I, is merely the par-
ticular case of this, when a = - •!
2 -■
Thus (cos a -I- \/ — 1 sina) (a + ^-s/— 1)
= a cos o — 5 sina-l- \/— 1 (asino + J cos a),
by direct multiplication. The reader will at once see that the new
form indicates that a rotation through an angle a has taken place,
if he compares it with the common formulae for turning the co-
ordinate axes through a given angle. Or, in a less simple manner,
thus —
Length =\/(a coso— 6sina)^ + (asina + 5cosa)^
= \/a'^ -I- b'^ as before.
12.] VECTORS, AND THEIR COMPOSITIO^^ 3
Inclination to axis of a; j
, tan a-\ —
, _, « sin a + cos a , , a
= tan '■ j-^ — = tan-i =
a cos a— sin a o
I 1 tana
= a + tan~i - •
a
8.] We see now, as it were, wA^ it happens that
(cos a 4- V — 1 sin a)™ = cos »ia + /^/^^^ sin ma.
In fact, the first operator produces m successive rotations in the
same direction, each through the angle a ; the second, a single
rotation through the angle ma.
9.] It may be interesting, at this stage, to anticipate so far as to
state that a Quaternion can, in general, he put under the form
N {cos d + -ay sin 6),
where iV" is a numerical quantity, 8 a real angle, and
This expression for a quaternion bears a very close analogy to the
forms employed in De Moivre's Theoreili ; but there is the essential
difference (to which Hamilton's chief invention referred) that -sr,
is not the algebraic v — 1, but may be an^ directed unit-line what-
ever in space.
10.] In the present century Argand, Warren, and others, extended
the results of WalHs and De Moivre. They attempted to express
as a line the product of two lines each represented by a symbol
such as a + J v^— 1. To a certain extent they succeeded, but sim-
plicity was not gained by their methods, as the terrible array of
radicals in Warren's Treatise suflBciently proves.
11.] A very curious speculation, due to Servois and published
in 1813 in Gergonne's Annates, is the only one, so far as has
been discovered, in which the slightest trace of an anticipation of
Quaternions is contained. Endeavouring to extend to space the
form a + J\/— 1 for the plane, he is guided by analogy to write for
a directed unit-line in space the form
p cos a + §■ cos /3 + r cos y,
where a, p, y are its inclinations to the three axes. He perceives
easily that j9, q, r must be non-reals : but, he asks, " seraient-elles
imaginaires reductibles a la forme generale A-\-B '^ — I ?" This
he could not answer. In fact they are the i, j, k of the Quaternion
Calculus. (See Chap. II.)
12.] Beyond this, few attempts were made, or at least recorded, in
earlier times, to extend the principle to space of three dimensions ;
B a
4 QUATERNIONS. [ 1 3-
and, though many such have been made within the last forty
years, none, with the single exception of Hamilton's, have
resulted in simple, practical methods; all, however ingenious,
seeming to lead at once to processes and results of fearful com-
plexity.
For a lucid, complete, and most impartial statement of the
claims of his predecessors in this field we refer to the Preface to
Hamilton's Lectures on Quaternions.
13.] It was reserved for Hamilton to discover the use of -n/ — 1
as a geometric realitij, tied down to no particular direction in space,
and this use was the foundation of the singularly elegant, yet
enormously powerful, Calculus of Quaternions.
While all other schemes for using -s/^ to indicate direction
make one direction in space expressible by real numbers, the re-
mainder being imaginaries of some kind, leading in general to
equations which are heterogeneous ; Hamilton makes all directions
in space equally imaginary, or rather equally real, thereby ensuring
to his Calculus the power' of dealing with space indifferently in
all directions.
In fact, as we shall see, the Quaternion method is independent
of axes or any supposed directions in space, and takes its reference
lines solely from the problem it is applied to.
14.] But, for the purpose of elementary exposition, it is best
to begin by assimilating it as closely as we can to the ordinary
Cartesian methods of Geometry of Three Dimensions, which are
in fact a mere particular case of Quaternions in which most of
the distinctive features are lost. We shall find in a little that
it is capable of soaring above these entirely, after having employed
them in its establishment; and, indeed, as the inventor's works
amply prove, it can be established, ah initio^ in various ways,
without even an allusion to Cartesian Geometry. As this work
is written for students acquainted with at least the elements of
the Cartesian method, we keep to the first-mentioned course of
exposition ; especially as we thereby avoid some reasoning which,
though rigorous and beautiful, might be apt, from its subtlety,
to prove repulsive to the beginner.
We commence, therefore, with some very elementary geometrical
ideas.
15.] Suppose we have two points A and B in space, and suppose
A given, on how many numbers does ^'s relative position depend ?
If we refer to Cartesian coordinates (rectangular or not) we find
1 9-] VECTOES, AND THEIR COMPOSITION. 5
•
that the data required are the excesses of ^'s three coordinates over
those of A. Hence three numbers are required.
Or we may take polar coordinates. To define the moon's position
with respect to the earth we must have its Geocentric Latitude
and Longitude, or its Right Ascension and Declination, and, in
addition, its distance or radius-vector. Three again.
16.J Here it is to be carefully noticed that nothing has been
said of the actual coordinates of either A or B, or of the earth
and moon, in space ; it is only the relative coordinates that are
contemplated.
Hence any expression, as AB, denoting a line considered with
reference to direction as well as length, contains implicitly three
numbers, and all lines parallel and equal to AB depend in the same
way upon the same three. Hence, all lines which are equal and
parallel may he represented hy a common symbol, and that symbol
contains three distinct numbers. In this sense a line is called a
VEOTOE, since by it we pass from the one extremity, A, to the
other, B ; and it may thus be considered as an instrument which
carries A to B : so that a vector may be employed to indicate a
definite translation in space.
17.] We may here remark, once for all, that in establishing a
new Calculus, we are at liberty to give any definitions whatever
of our symbols, provided that no two. of these interfere with, or
contradict, each other, and in doing so in Quaternions simplicity
and (so to speak) naturalness were the inventor's aim.
18.] Let AB be represented by a, we know that a depends on
three separate numbers. Now if CD be equal in length to AB
and if these lines be parallel, we have evidently CI) = AB = a,
where it will be seen that the sign of equality, between vectors
contains implicitly equality in length and parallelism in direction.
So far we have extended the meaning of an algebraical symbol.
And it is to be noticed that an equation between vectors, as
a = /3,
contains three distinct equations between mere numbers.
19.] We must now define + (and the meaning of — will follow)
in the new Calculus. Let A, B, C be any three points, and (with
the above meaning of =) let
AB=a, BG=I3, AC=y.
If we define + (in accordance with the idea (§ 16) that a vector
represents a translation) by the equation
6 QUATERNIONS. [20.
or AB + £C = AC,
we contradict nothing that precedes, but we at once introduce the
idea that vectors areata he compounded, in direction and magnitude,
like simultaneous velocities. A reason for this may be seen in
another way if we remember that by adding the diflferences of the
Cartesian coordinates of A and B, to those of the coordinates of
B and C, we get those of the coordinates of A and C. Hence these
coordinates enter" linearly into the expression for a vector.
20.] But we also see that if C and A coincide (and C may be
any point) AQ = 0,
for no vector is then required to carry A to C. Hence the above
relation may be written, in this case,
AB+BA = 0,
or, introducing, and by the same act defining, the symbol — ,
BA=-AB.
Hence, t/ie symbol — , applied to a vector, simply shows that its
direction is to he reversed.
And this is consistent with all that precedes ; for instance,
ab+bc = m;,
and AB = AC-BC,
or =AG+CB,
are evidently but different expressions of the same truth.
21.] In any triangle, ABC, we have, of course,
IB + BC+CA^ 0;
and, in any closed polygon, whether plane or gauche,
AB-k^BC+ '. + TZ+ZA = 0.
In the case of the polygon we have also
AB + BC+ + fZ=AZ.
These are the well-known propositions regarding composition of
velocities, which, by the second law of motion, give us the geo-
metrical laws of composition of forces.
22.] If we compound any number of parallel vectors, the result
is obviously a numerical multiple of any one of them.
Thus, if A, B, C are in one straight line,
BC=i>!AB;
where a; is a number, positive when B lies between A and C, other-
wise negative : but such that its numerical value, independent
of sign, is the ratio of the length of BC to that of AB. This is
25-] VECTOES, AND THEIR COMPOSITION. 7
at oace evident if AB and BC be commensurable j and is easily
extended to incommensurables by the usual reductio ad absurdum.
23.] An important, but almost obvious, proposition is that any
vector may he resolved, and in one way only, into three components
parallel respectively to any three given vectors, no two of which are
parallel, and which are not parallel to one plane.
Let OA, OB, OC be the three fixed vectors, c
OP any other vector. From P draw PQ
parallel to CO, meeting the plane BOA in Q.
[There must be a definite point Q, else PQ,
and therefore CO, would be parallel to BOA,
a case specially excepted.] Prom Q draw
QB parallel to BO, meeting OA in B. Then
we have OP = 0^ + ^ + QP (§ 21),
and these components are respectively parallel to the three given
vectors. By § 22 we may express OB as a numerical multiple
of OA, RQ oi OB, and QP of OC. Hence we have, generally, for
any vector in terms of three fixed non-coplanar vectors, a, /3, y,
OP = p = xa + yl3 + zy,
which exhibits, in one form, the three numbers on which a vector
depends (§ 16). Here x, y, z are perfectly definite, and can have
but single values.
24.] Similarly any vector, as OQ, in the same plane with OA
and OB, can be resolved into components OB, RQ, parallel re-
spectively to OA and OB ; so long, at least, as these two vectors
are not parallel to each other.
25.] There is particular advantage, in certain cases, in employ-
ing a series of three mutually perpendicular unit-vectors as lines of
reference. This system Hamilton denotes by i,j, h.
Any other vector is then expressible as
p = xi-\-yj-\-zh.
Since i, j, k are unit- vectors, x, y, z are here the lengths of con-
terminous edges of a rectangular parallelepiped of which p is the
vector-diagonal ; so that the length of p is, in this case,
Let TO- = ^i + T/y+C^
be any other vector, then (by the proposition of § 23) the vector
equation p =. 'ss
obviously involves the following three equations among numbers,
« = ^. y = ■<), z=C
8 QUATERNIONS. [26.
Suppose i to be drawn eastwards, J northwards, and k upwards,
this is equivalent merely to saying that if two points coincide, ihey
are equally to tie east {or west) of any third point, equally to the
north {or south) of it, and equally elevated above {or depressed below)
its level.
26.] It is to be carefully noticed that it is only when a, fi, y are
not coplanar that a vector equation such as
p = OT,
or «a-f ^;3 + «y = £o + jj/3+Cy,
necessitates the three numerical equations
m = i, y = n, « = ^
For, if a, ^j y be coplanar (§ 24), a condition of the following form
must hold y = aa + b^.
Hence p = {x + za)a+{y + zh)p,
^={i+Ca)a + {r, + Cb)^,
and the equation p ■= m
now requires only the two numerical conditions
x + za= ^+Ca, y + zb = r] + (b.
27.] The Commutative and Associative Laws hold in the combination
of vectors by the signs + and — . It is obvious that, if we prove
this for the sign + , it will be equally proved for — , because —
before a vector (§ 20) merely indicates that it is to be reversed
before being considered positive.
Let A, B, G, B be, in order, the corners of a parallelogram ; we
have, obviously, Jb = SC, IT) = BG.
And AB + BC= IC=An+BC=BC+AB.
Hence the commutative law is true for the addition of any two
vectors, and is therefore generally true.
Again, whatever four points are represented by A, B, C, J), we
have lD = IB+BB = AC-\-UD,
or substituting their values for AB, BB, AC respectively, in these
three expressions,
lB+BC-\-CB^AB + {BC+CB)= {AB + BC) + CB.
And thus the truth of the associative law is evident.
28.] The equation „ — xB,
where p is the vector connecting a variable point with the origin,
/3 a definite vector, and x an indefinite number, represents the
straight line drawn from the origin parallel to ^ (§ 22).
30.J VECTOES, AND THEIR COMPOSITION. 9
The straight line drawn from A, where OA = a, and parallel
to j8, has the equation
p = a + a;/3 ; (1)
In words, we may pass directly from to P by the vector OP or p ;
or we may pass first to A, by means of OA or a, and then to P
along a vector parallel to ^ (§ 16).
Equation (1) is one of the many useful forms into which Quater-
nions enable us to throw the general equation of a straight line in
space. As we have seen (§ 25) it is equivalent to three numerical
equations ; butj as these involve the indefinite quantity x, they are
virtually equivalent to but two, as in ordinary Geometry of Three
Dimensions.
29.] A good illustration of this remark is furnished by the fact
that the equation p = va + se^,
which contains two indefinite quantities^ is virtually equivalent to
only one numerical equation. And it is easy to see that it re-
presents the plane in which the lines o and fi lie ; or the surface
which is formed by drawing, through every point of OA, a line
parallel to OB. In fact, the equation, as written, is simply § 24
in symbols.
And it is evident that the equation
P = y+ya + oo^
is the equation of a plane passing through the extremity of y, and
parallel to a and ;8.
It will now be obvious to the reader that the equation
P =i'i«i+i'2a2+ = '2pa,
where a^, Og, &c. are given vectors, and Pi,P2> ^c. numerical quan-
tities, represents a strd/igM line if i5i,j»2J &c. be linear functions of
one indeterminate number ; and a plane, if they be linear expres-
sions containing two indeterminate numbers. Later (§31 {})), this
theorem will be much extended.
Again, the equation p = xa + y^-^zy
refers to any point whatever in space, provided a, /3, y are not
coplanar. {Ante, § 23).
30.] The equation of the line joining any two points A and B,
where OA = a and OB = 13, is obviously
P = a + a;(/3— a),
or p = ^ + y(a-/3).
These equations are of course identical, as may be seen by putting
1—y for ss.
10 QUATERNIONS. [3 1.
The first may be written
p + (x—l)a — x^ = ;
or j)p + qa + rfi = 0,
subject to the condition p + q + r = identically. That is — A
homogeneous linear function of three vectors, equated to zero,
expresses that the extremities of these vectors are in one straight
line, if the sum of the coefficients he identically zero.
Similarly, the equation of the plane containing the extremities
A, B, C of the three non-coplanar vectors a, j3, y is
p = a + a:(/3-a) + y(y-/3),
where x and y are each indeterminate.
This may be written
pp + qa + r^ + sy = 0,
with the identical relation
p + q + r + s = 0.
which is the condition that four points may lie in one plane.
31.] We have already the means of proving, in a very simple
manner, numerous classes of propositions in plane and solid geo-
metry. A very few examples, however, must suflSce at this stage ;
since we have hardly, as yet, crossed the threshold of the subject,
and are dealing with mere linear equations connecting two or more
vectors, and even with them we are restricted as yet to operations of
mere addition. We will give these examples with a painful minute-
ness of detail, which the reader will soon find to be necessary only
for a short time, if at all.
(a.) The diagonals of a parallelogram bisect each other.
Let ABGB be the parallelogram, the point of intersection of
its diagonals. Then
iO + ^= IB =^G = Bb+OC,
which gives AO-OC = BO-OB.
The two vectors here equated are parallel to the diagonals respect-
ively. Such an equation is, of course, absurd unless
(1) The diagonals are parallel, in which case the figure
is not a parallelogram ;
(2) Jo = Oa, and ^ = OB, the proposition.
(i.) To show that a triangle can he constructed, whose sides
are parallel, and equal, to the hisectors of the sides of
any triangle.
Let ABC be any triangle, Aa, Bh, Cc the bisectors of the sides.
3I-] VECTORS, AND THEIR COMPOSITION. 11
_•
Then Aa, = AB+Ba = AB^-\ BC,
Bb - - - =BC+hCA,
Co - - - =Cl+\lB.
Hence Aa + Bb + Cc=^{lB + BG+CA)^();
which (§21) proves the proposition.
Also Aa — JS+\BC
= AB-\{Cl+AB)
= ^{AB-ai) = i{lB+IC),
results which are sometimes useful. They may be easily verified
by producing Aa to twice its length and joining the extremity
with B.
{¥.) The bisectors of the sides of a triangle meet in a point, which
trisects each of them.
Taking A as origin, and putting o, /3, y for vectors parallel, and
equal, to the sides taken in order BC, CA, AB; the equation of
Bb is (§ 28 (1))
p = y + «(y+f) = (i+a')y + |^-
That of Cc is, in the same way,
p = -(l + y)^-|y.
At the point 0, where Bb and Cc intersect,
p=(H-a;)y+-/3=-(l+j.)^-|y.
Since y and ^ are not parallel, this equation gives
H-«' = -f, and | = _(i+y).
From these a; = y = — |.
Hence iO = 4 (y-/3) = 1 2a. (See Ex. («).)
This equation shows, being a vector one, that Aa passes through 0,
and that JO : Oa : : 2: 1.
(c) If 02 = a, ^
OG=aa+b^,
be three given co-planar
«^
^
^^"^^^ /
\
^^^/s^~-~^
vectors, and the lines in-
^^
fli
dicated in the figure be drawn, the points «i,*i,Ci lie in a straight
line.
12
QUATERNIONS.
[31.
We see at once, by the process indicated in § 30, that
Oc =
aa + b^ Qj ^ "-"■
a + b
Hence we easily find
-a — 2b ^
0-a= '^
\—a
Oa^ =■
Oc, = — 7
^ b—a
\—a-2b •■ l — 2a-b'
These give
-{l-a-2b)Oai+{l-2a-6)Obi-{b-a)Oc^ = 0.
But ■ _(l-a-2i) + (l-2a-5)-(5-a) = identically.
This, by § 30, proves the proposition.
(d.) Let 02 = a, OB = /3, be any two vectors. If MP be
parallel to OB; and OQ, BQ, be drawn parallel to AP,
OP respectively; the locus of Q is a straight line parallel
to OA.
Let OM = ea.
Then_
AP = e— la + a;)3.
Hence the equation of
OQis
p = y(e-la+»i3);
and that of ^Q is
p = ^ + z{ea+co^).
At Q we have, therefore,
xy = l+zx, \
y{e-\) = ze.\
These give xy = e, and the equation of the locus of Q is
P = «/3 + /a,
i. e. a straight line parallel to OA, drawn through N in OB pro-
duced, so that ON -.OB:: OM: OA.
CoE. li BQ meet MP in q,P'c[=^; and if AP meet NQ in p,
Qp=a. _ _
Also, for the point B we have pB = AP, Q,R=Bq.
Hence, if from any two points, A and B, lines be drawn intercepting
a given length Pq on a given line Mq; and if, from B their point of
intersection, Bp be laid off = PA, and BQ = qB ; Q and p lie on a
fixed straight line, and the length of Qp is constant.
(e.) To find the centre of inertia of any system.
If OA = a, OB = a^, be the vector sides of any triangle, the
vector from the vertex dividing the base AB in C so that
31.]
VECTORS, AND THEIE COMPOSITION.
13
BG : CA ■.:m:m-^ is
For A£ is Oj — a, and therefore AC is
-(oj— a).
Hence
00:
OA + AC
= a +
(«! — a)
.,i + Ml
ma + mj^a^
This expression shows how to find the centre of inertia of two
masses; m at the extremity of a, m^ at that of a^. Introduce Wg
at the extremity of 02, then the vector of the centre of inertia of the
three is, by a second application of the formula,
»i4-««j
M a + »Zi Oj^ + »«2 02
■m-\-m■^^-\-m^
(m+mj^ + m^
For any number of masses, expressed generally by m at the extre-
mity of the vector a, we have the vector of the centre of inertia
'^ s(ot) ■
This may be written 2m(a—fi) = 0.
Now Oj— /3 is the vector of % with respect to the centre of inertia.
Hence the theorem, ^ the vector of each element of a mass, drawn
from the centre of inertia, he increased in length in proportion to the
mass of the element, the sum of all these vectors is zero.
(_/.) We see at once that
the equation
where t is an indeterminate
number, and a, j8 given vec-
tors, represents a parabola.
The origin, 0, is a point on
the curve, /3 is parallel to
the axis, i. e. is the diameter
OB drawn from the origin,
and a is OA the tangent at the origin.
qp = at, 6q =
14 QUATERNIONS. [31.
The secant joining the points where t has the values t and If is
represented by the equation
, = a. + ^+.(ar+^-a.-^) (§30)
Put 1f=.t, and write x for x{if—t) [which may have any value]
and the equation of the tangent at the point {t) is
Put X = —t, p = — >
or the intercept of the tangent on the diameter is —the abscissa of
the point of contact.
Otherwise: the tangent is parallel to the vector a+fit or
at + pt'' OT at + if. + if. ov 0Q+ UP. BuifF=fd + 6P,
hence TO = OQ.
•{ff.) Since the equation of any tangent to the parabola is
p = at + ^ + x{a+lii),
let us find the tangents which can be drawn from a given point.
Let the vector of the point be
p=pa + ql3 (§24).
Since the tangent is to pass through this point, we have, as con-
ditions to determine t and x, i + x = p,
-j + xt = q;
by equating respectively the coefficients of a and /3.
Hence ^ =jo+ \/^^ — 2$'.
ThuSj in general, two tangents can be drawn from a given point.
These coincide if ^2 _ 2^ ;
that is, if the vector of the point from which they are to be drawn
is „ P^ „
p =pa + qfi =Pa.+ ^p,
i. e. if the point lies on the parabola. They are imaginary if
2q>p^, i. e. if the point be
r being positive. Such a point is evidently within the curve, as at
72, where OQ =^/3, QP=pa, PB = r^.
3I-] VECTORS, AND THEIB COMPOSITION. 15
(A.) Calling the values o{ t f9r the two tangents found in (^)
ti and ^2 respectively, it is obvious that the vector join-
ing the points of contact is
"2
2
which is parallel to f ^^
a + ^-i^;
or, by the values of t^ and t^ in (ff),
a+jB/3.
Its direction, therefore, does not depend on q. In words, If pairs
of tangents he ckamn to a parabola from points of a diameter pro&uced,
the chords of contact are parallel to the tangent at the vertex of the
diameter. This is also proved by a former result, for we must have
OT for each tangent equal to Q 0.
{i.) The equation of the chord of contact, for the point whose
vector is p=pa + ql3,
Bt ^
is thus P = a^i+^ + ^(«+J0;8).
Suppose this to pass always through the point whose vector is
p = aa + b^.
Then we must have , ,
h+^ = «. )
t ^ [
or ti=p±^p^ — 2pa + 2b.
Comparing this with the expression in {g), we have
q =pa—h;
that is, the point from which the tangents are drawn has the vector
p =pa + {pa—b)^
= —b^+p{a + aj3), a straight line (§ 28 (1)).
The mere form of this expression contains the proof of the usual
properties of the pole and polar in the parabola ; but, for the sake
of the beginner, we adopt a simpler, though equally general,
process.
Suppose a = 0. This merely restricts the pole to the particular
diameter to which we have referred the parabola. Then the pole
is Q, where p = b^;
and the polar is the line TU, for which
p = -6fi+pa.
16 QTTATEKNIONS. [3I.
Hence the polar of any point is parallel to the tangent at the extremity
of the diameter on which the point lies, and its intersection with that
diameter is as far beyond the vertex as the pole is within, and vice
versa.
(J.) As another example let us prove the following theorem.
Jf a triangle he inscribed in a parabola, tlie three points
in which the sides are met by tangents at the angles lie in
a straight line.
Since is any point of the curve, we may take it as one corner
of the triangle. Let t and ^j determine the others. Then, if
OTj, OTj, iirg represent the vectors of the points of intersection of the
tangents with the sides, we easily find
tt.
''' = t:vt""
These values give
itj^-t 2t—t-i^ t^-t^
Ai "^h-t ^t-h ty-^ „ •. ^. 1,
Also — i — -i- — i- — = identically.
Hencoj by § 30, the proposition is proved.
ijc) Other interesting examples of this method of treating
curves will, of course, suggest themselves to the
student. Thus
p = a cos if 4- ^ sin ^
or p = oa;+^^/l— jc^
represents an ellipse, of which the given vectors a and /3 are semi-
conjugate diameters.
Agam, p = aif + - or p = a tana;+^cota;
evidently represents a hyperbola referred to its asymptotes.
But, so far as we have yet gone with the explanation of the
calculus, as we are not prepared to determine the lengths or in-
clinations of vectors, we can investigate only a very small class of
the properties of curves, represented by such equations as those
above written.
33-] VECTORS, AND THEIR COMPOSITION. 17
(I.) We may now, in extensi^ of the statement in § 29, make
the obvious remark that
p = Sj)a
is the equation of a curve in space, if the numbers i'i,i»2> ^^- ^^^
functions of one indeterminate. In such a case the equation is
sometimes written . _ j^/f.
But, if jOi, j»2J ^c. be functions of two indeterminates, the locus of
the extremity of p is a surface ; whose equation is sometimes written
p = <t>{t,u).
(m.) Thus the equation
p = acost + ^sint+yt
belongs to a helix.
Again, p=pa + q^ + ry
with a condition of the form
ap^ 4 6q^ + cr^ ■= I
belongs to a central surface of the second order, of which a, p, y
are the directions of conjugate diajneters. If a, b, c be all positive,
the surface is an ellipsoid.
32.] In Example (_/) above we performed an operation equi-
valent to the differentiation of a vector with reference to a single
numerical variable of which it was given as an explicit function.
"As this process is of very great use, especially in quaternion investi-
gations connected with the motion of a particle or point ; and as it
will afford us an opportunity of making a preliminary step towards
overcoming the novel difficulties which arise in quaternion differen-
tiation; we will devote a few sections to a more careful exposition
of it.
33.] It is a striking circumstance, when we consider the way
in which Newton's original methods in the Differential Calculus
have been decried, to find that Hamilton was obliged to employ
them, and not the more modern forms, in order to overcome the
characteristic difficulties of quaternion differentiation. Such a thing
as a differential coefficient has absolutely no meaning in quaternions,
except in those special cases in which we are dealing with degraded
quaternions, such as nvmibers, Cartesian coordinates, &c. But a
quaternion expression has always a differential, which is, simply,
what Newton called sifluadon.
As with the Laws of Motion, the basis of Dynamics, so with the
foundations of the Differential Calculus ; we are gradually coming
to the conclusion that Newton's system is the best after all.
c
18 QtTATERNIONS. [34-
34.J Suppose p to be the vector of a curve in space. Then,
generally, p may be expressed as the sum of a number of terms,
each of which is a multiple of a given vector by a function of some
one indeterminate; or, as in § 31 (1), if P be a point on the curve,
6P=p = 4>{t).
And, similarly, if Q be ani/ other point on the curve,
where htis any number whatever.
The vector-chord PQis therefore, rigorously,
6p = pi-p = (f>{t + bt)-cl>t.
35.] It is obvious that, in the present case, because the vectors
involved in (j) are constant, and their numerical multipliers alone vary,
the expression i^it^ht) is, by Taylor's Theorem equivalent to
'^^*^^-ir^*^~d(^~^^
Hence, ^<^(0 ,,^^'<^ W (8^)' ^^
And we are thus entitled to write, when ht has been made inde-
finitely small, , , ,,.
^ ' ,lp. dp dc\,{t) _ ,
In such a case as this, then, we are permitted to differentiate,
or to form the differential coefficient of, a vector, according to the
ordinary rules of the Differential Calculus. But great additional
insight into the process is gained by applying Newton's method.
36.] Let OP be
_ P = <i>[t),
and OQi
p= 4>{t + dt),
where dt is any number whatever.
The number t may here be taken
as representing time, i. e. we may
suppose a point to move along the
curve in such a way that the value
of t for the vector of point P of the
curve denotes the interval which has
elapsed (since a fixed epoch) when the moving point has reached
the extremity of that vector. If, then, dt represent any interval,
finite or not, we see that
6q^=^{t+dt)
will be the vector of the point after the additional interval dt.
38.J YECTORS, AND THEIR COMPOSITION. 19
But this, in general, gives us little or no information as to the
velocity of the point at P. We shall get a better approximation
by halving the interval di, and finding Q^, where 0^2= <^ {i+h ^i))
as the position of the moving point at that time. Here the vector
virtually described in ^df is PQ^. To find, on this supposition,
the vector described in di, we must double PQ2) and we find, as a
second approximation to the vector which the moving point would
have described in time dt, if it had moved for that period in the
direction and with the velocity it had at P,
Tq2=2PQ^ = 2{0Q2-6P)
= 2{(l>{i+kdt)-(l){t)}.
The next approximation gives
P^, = 3PQ,= 3{6Q,-6P)
= 3{4>{i+idt)-<t>{i)].
And so on, each step evidently leading us nearer the sought truth.
Hence, to find the vector which would have been described in time
dt had the circumstances of the motion at P remained undisturbed,
we must find the value of
dp = Tq = J^:,=^ai^cj>(t + ^dt)-<j>{t)\-
We have seen that in this particular case we may use Taylor's
Theorem. We have, therefore,
dp = J^,=^ X \^'{t)\ di+<j>"{t) ^ ^ 4 &c. I
= 4)' (t) dt.
And, if we choose, we may now write
37.] But it is to be most particularly remarked that in the
whole of this investigation no regard whatever has been paid to
the magnitude of dt. The question which we have now answered
may be put in the form — A point describes a given curve in a given
manner. At any point of its path its motion suddenly ceases to he
accelerated. What space will it describe in a definite interval ? As
Hamilton well observes, this is, for a planet or comet, the case
of a ' celestial Atwood's machine.'
38.] If we suppose the variable, in terms of which p is expressed,
to be the arc, s, of the curve measured from some fixed point, we
find as before , ,,,,, ,, ^'{t)ds
dp = ^{t)dt = ^^-^
= 4>'{s}ds. ^^
C 3
20 QUATERNIONS. [39.
From the very nature of the question it is obvious that the length
of dp must in this case 'be ds. This remark is of importance, as
we shall see later ; and it may therefore be useful to obtain afresh
the above result without any reference to tiine or velocity.
39.] Following strictly the process of Newton's Vllth Lemma,
let us describe on Pq^ an arc similar to PQg, and so on. Then
obviously, as the subdivision of ds is carried farther, the new arc
(whose length is always ds) more and more nearly coincides with
the line which expresses the corresponding approximation to <?p.
40.] As a final example let us take the hyperbola
Here dp = {a-^)dt.
This shews that the tangent is parallel to the vector
at --•
In words, if the vector {from the centre) of a point in a hyperbola
he one diagonal of a parallelogram, two of whose sides coincide with
the asymptotes, the other diagonal is parallel to the tangent at the
point.
41.] Let us reverse this question, and seek the envelope of a line
which cuts off from two fixed axes a triangle of constant area.
If the axes be in the directions of a and fi, the intercepts may
evidently be written at and y . Hence the equation of the line is
(§30)
p = at-\-x{Y—aty
The condition of envelopment is, obviously, (see Chap. IX.)
dp = 0.
This gives =\a-x{^ + a)\dt+ {^-at)dx*.
Hence {\—x)dt — tdx=0,
J X ^, dx ^
and — — dt+ -^ = 0.
* We are not here to equate to zero the coefficients of dt and dx; for we must
remember that this equation is of the form
=pa + q$,
where p and q are numbers ; and that, so long as a and are actual and non-parallel
vectors, the existence of such an equation requires
i> = 0, 5 = 0.
43- J VECTORS, AND THEIK COMPOSITION. 21
From these, at once, x = \, since dx and dt are indeterminate.
Thus the equation of the envelope is
the hyperbola as before ; a, ;3 being portions of its asymptotes.
42.] It may assist the student to a thorough comprehension
of the above process, if we put it in a slightly different form.
Thus the equation of the enveloping line may be written
p = ai!(l-a!) + /3*,
which gives dp = = ad {t (1 —x))+^d (-) •
Hence, as a is not parallel to /3, we must have
d{t{l-x)) = (i, ^(f) = 0;
and these are, when expanded, the equations we obtained in the
preceding section.
43.] For farther illustration we give a solution not directly em-
ploying the differential calculus. The equations of any two of the
enveloping lines are
p = at + X 0-: at\t
p =aifi + a?i(— -ai5i)>
t and <i being given, while x and ajj are indeterminate.
At the point of intersection of these lines we have (§ 26),
t{l-x) = ^i(l-«i), \
X _Xi >
These give, by eliminating x^,
t{\-x) = ti{l-^x),
t
or X =■ . •
ti + t
Hence the vector of the point of intersection is
22 QUATERNIONS. [44.
and thus, for the ultimate intersections, where ^^ = 1,
p = ^ (a^ + y) as before.
CoE. (1). If tt^ = 1,
a + /3 .
or the intersection lies in the diagonal of the parallelogram on a, j3.
Cob. (2). If ti = mt, where m is constant,
mta + —
V
P = ■
But we have also iv =
m+ 1
1
i+l
Hence tAe locus of a point which divides in a given ratio a line
cutting off a given area from, two fixed axes, is a hyperbola of which
these axes are the asymptotes.
Cor. (3). If we take
tt^ (^+ ^1) = constant
the locus is a parabola ; and so on.
44.] The reader who is fond of Anharmonic Ratios and Trans-
versals will find in the early chapters of Hamilton's Elements of
Quaternions an admirable application of the composition of vectors
to these subjects. The Theory of Geometrical Nets, in a plane,
and in space, is there very fully developed ; and the method is
shewn to include, as particular cases, the processes of Grassmann's
Ausdehnungslehre and Mobius' Barycentrische Calcul. Some very
curious investigations connected with curves and surfaces of the
second and third orders are also there founded upon the composition
of vectors.
EXAMPLES TO CHAPTER I.
1. The lines which join, towards the same parts, the extremities
of two equal and parallel lines are themselves equal and parallel.
{Euclid, I. xxxiii.)
2. Find the vector of the middle point of the line which joins
EXAMPLES TO CHAPTER I. 23
the middle poiats of the diagonals of any quadrilateral, plane or
gauche^ the vectors of the corners being given ; and so prove that
this point is the mean point of the quadrilateral.
If two opposite sides be divided proportionally, and two new
quadrilaterals be formed by joining the points of division, the mean
points of the three quadrilaterals lie in a straight line.
Shew that the mean point may also be found by bisecting the
line joining the middle points of a pair of opposite sides.
3. Verify that the property of the coefficients of three vectors
whose extremities are in a line (§ 30) is not interfered with by
altering the origin.
4. If two triangles ABC, abc, be so situated in space that Aa,
Bb, Cc meet in a point, the intersections of AB, ah, of BG, be, and
of CA, ca, lie in a straight line.
5. Prove the converse of 4, i. e. if lines be drawn, one in each
of two planes, from any three points in the straight line in which
these planes meet, the two triangles thus formed are sections of
a common pyramid.
6. If five quadrilaterals be formed by omitting in succession each
of the sides of any pentagon, the lines bisecting the diagonals of
these quadrilaterals meet in a point. (H. Fox Talbot.)
7. Assuming, as. in § 7, that the operator
cos 6 + \/— 1 sin 6
turns any radius of a given circle through an angle 6 in the
positive direction of rotation, without altering its length, deduce
the ordinary formulae for cos [A + B), cos {A—B), sin {A + B), and
sin [A—B), in terms of sines and cosines of A and B.
8. If two tangents be drawn to a hyperbola, the line joining
the centre with their point of intersection bisects the lines joining
the points where the tangents meet the asymptotes : and the
tangent at the point where it meets the curves bisects the intercepts
of the asymptotes.
9. Any two tangents, limited by the asymptotes, divide each
other proportionally.
10. If a chord of a hyperbola be one diagonal of a parallelogram
whose sides are parallel to the asymptotes, the other diagonal passes
through the centre.
11. Shewthat p = x^ a + f ^-\-{x-\-yf y
is the equation of a cone of the second degree, and that its section
by the plane _ pa + g^ + ry
^~ p + q+r
24 QUATERNIONS.
is an ellipse which touches, at their middle points, the sides of
the triangle of whose corners a, /3, y are the vectors. (Hamilton,
Elements, p. 96.)
12. The lines which divide, proportionally, the pairs of opposite
sides of a gauche quadrilateral, are the generating lines of a hyper-
bolic paraboloid. (Ibid. p. 97.)
13. Shew that p = x^a + y^fi + z^y,
where x + y + z = 0,
represents a cone of the third order, and that its section by the plane
'' ~ p + q + r
is a cubic curve, of which the lines
P + 1
are the asymptotes and the three (real) tangents of inflexion. Also
that the mean point of the triangle formed by these lines is a
conjugate point of the curve. Hence that the vector a-f-(3 + y is a
conjugate ray of the cone. (Ibid. p. 96.)
CHAPTER ir.
PRODUCTS AND QUOTIENTS OF VECTOES.
45.] We now come to the consideration of points in which the
Calculus of Quaternions differs entirely from any previous mathe-
matical method ; and here we shall get an idea of what a Qua-
ternion is, and whence it derives its name. These points are
fundamentally, involved in the novel use of the symbols of mul-
tiplication and division. And the simplest introduction to the
subject seems to be the consideration of the quotient, or ratio, of
two vectors.
46.] If the given vectors be parallel to each other, we have
already seen (§ 22) that either may be expressed as a numerical
multiple of the other; the multiplier being simply the ratio of
their lengths, taken positively if they are similarly directed, nega-
tively if they run opposite ways.
47.] If they be not parallel, let OA and OB be drawn parallel
and equal to them from any point ; and the question is reduced
to finding the value of the ratio of two vectors drawn from the
same point. Let us try to find upon how many distinct numbers this
ratio depends.
We may suppose OA to be changed into 0£ by the following
processes.
1st. Increase or diminish the length of OA till it becomes
equal to that of OB. For this only one number is
required, viz. the ratio of the lengths of the two
vectors. As Hamilton remarks, this is a positive, or
rather a signless, number.
2nd. Turn OA about until its direction coincides with that
of OB, and (remembering the effect of the first operation)
26 QUATERNIONS. [48.
we see that the two vectors now coincide or become
identical. To specify this operation three more numbers
are required, viz. two angles (such as node and inclina-
tion in the case of a planeVs orbit) to fix the plane in
which the rotation takes place, and one angle for the
amount of this rotation.
Thus it appears that the ratio of two vectors, or the multiplier
required to change one vector into another, in general depends upon
four distinct numbers, whence the name quaternion.
The particular case of perpendicularity of the two vectors, where
their quotient is a vector perpendicular to their plane, is fully con-
sidered below ; §§ 64, 65, 72, &c.
48.] It is obvious that the operations just described may be
performed, with the same result, in the opposite order, being per-
fectly independent of each other. Thus it appears that a quaternion,
considered as the factor or agent which changes one definite vector
into another, may itself be decompofed into two factors of which
the order is immaterial.
The stretching factor, or that which performs the first operation
in § 47, is called the Tensou, and is denoted by prefixing T to the
quaternion considered.
The turninff factor, or that corresponding to the second operation
in § 47, is called the Versoe, and is denoted by the letter U prefixed
to the quaternion.
49.] Thus, if OA = a, OB = fi, and if q be the quaternion which
changes a to /3, we have
13 = qa,
which we may write in the form
— = q, or ^a-i = q,
a
if we agree to defne that
— .a = |3a-i. o = p.
Here it is to be particularly noticed that we write q before a to
signify that a is multiplied by q, not q multiplied by a.
This remark is of extreme importance in quaternions, for, as we
shall soon see, the Commutative Law does not generally apply to
the factors of a product.
We have also, by §§ 47, 48,
q=TqUq=UqTq,
51.] PRODUCTS AND QUOTIENTS OF VECTORS. 27
where, as before, Tq^ depends merely on the relative lengths of
a and j3, and Uq depends solely on their directions.
Thus, if Oi and jSj be vectors of unit length parallel to a and j3
respectively, ^^^^^ u^^- = U^.
As will soon be shewn, when a is perpendicular to ^, the versor of
the quotient is quadrantal, i. e. it is a unit-vector.
50.] We must now carefully notice that the quaternion which
is the quotient when /3 is divided by a in no way depends upon
i\^e- absolute lengths, or directions, of these vectors. Its value
will remain unchanged if we substitute for them any other pair
of vectors which
(1) have their lengths in the same ratio,
(2) have their common plane the same or parallel,
and (3) make the same angle with each other.
Thus in the annexed figure
6^1^ OB
0^1 " OA
if, and only if,
^^^ O.Aj^ OA
(2) plane AOS parallel to plane A^O^B^,
(3) I.AOB = LA^O^B^.
[Equality of angles is understood to include
similarity in direction. Thus the rotation about
an upward axis is negative (or right-handed)
from OA to OB, and also from Oj A^ to 0^ B^r\
51.1 The Reciprocal of a quaternion q is defined by the equation,
1 -1 1
q- = qi ^=1.
Hence if - = ?> ov
a
^=qa,
a 1 J
we must have 'B~a ~ ^ '
a ■,
For this gives -.p = q '^.qa,
and each member of the equation is evidently equal to a.
^^/A
28 QUATERNIONS. [52.
Or, we may reason thus, q changes 61 to 0£, q-^ must therefore
change OB to OA, and is therefore expressed by - (§ 49).
The tensor of the reciprocal of a quaternion is therefore the
reciprocal of the tensor ; and the versor differs merely by the reversal
of its representative angle. The versor, it must be remembered,
gives the plane and angle of the turning — ^it has nothing to do
with the extension.
52.] The Conjugate of a quaternion q, written Kq, has the same
tensor, plane, and angle, only the angle is taken the reverse way.
Thus, if OA, OB, OA', lie in one plane, and if
0A'= OA, and LA:0B=IA0B, we have
OB ,6b ■ . n TT
-z=^ = a, and-^=- = coniugate 01 q ■=■ Kq.
OA ^' OA' ^^ ^ ^
By last section we see that
Kq = {Tqfq-\
Hence qKq = Kq.q = {Tqf.
This proposition is obvious, if we recollect that the
" tensors of q and Kq are equal, and -that the versors
are such that either annuls the effect of the other. The joint effect
of these factors is therefore merely to multiply twice over by the
common tensor.
53.] It is evident from the results of § 50 that, if a and ^ be
of equal length, their quaternion quotient becomes a versor (the
tensor being unity) and may be represented indifferently by any
one of an infinite number of ares of given length lying on the
circumference of a circle, of which the two vectors are radii. This
js of considerable importance in the proofs which follow.
fyo
Thus the versor ^=^ may be represented
in magnitude, plane, and direction (§ 50)
by the arc AB, which may in this extended
sense be written AB.
And, similarly, the versor ' is repre-
sented by A^B^ which is equal to (and
measured in the same direction as) AB if
jLAiOBi = LAOB,
i.e. if the versors are equal, in the quaternion meaning of the word.
56. j PRODUCTS AKD QTJOTIEN.TS OF VECTORS. 29
54.] By the aid of this process,''Vhen a versor is represented as
an arc of a great circle on the unit-sphere, we can easily prove that
qiiaternion multiplication is not generally commutative.
Thus let q be the versor j!b or ^=- •
^ ^ OA
Make BC = AB, (which, it must be
remembered, makes the points A, B, C
lie in one great circle), then q^ may also
be represented by •
^ OB
In the same way any other versor r
may be represented by BB or BB and bv -=- or -^=- •
^ OB OB
The line OB in the figure is definite, and is given by the inter-
section of the planes of the two versors ; being the centre of the
unit-sphere.
Now rOB = OB, and qOB = 00,
Hence _ qrOB=6c,
00 '-^
or qr = -=■ > and may therefore be represented by the arc BC of
a great circle.
But rq is easily seen to be represented by the arc AB.
For q02 = OB, and rOB = OB,
— OB
whence rq OA = OB, and rq = -=- •
OA
Thus the versors rq and qr, though represented by arcs of equal
length, are not generally in the same plane and are therefore un-
equal : unless the planes of q and r coincide.
Calling OA a, we see that we have assumed, or defined, in the
above proof, that q.ra = qr.a and r.qa = r'q.a when qa, ra, q.ra, and
r.qa are all vectors.
55.J Obviously CB is Kq, BB is Kr, and CB is K(qr). But
CB = BB.CB, which gives us the very important theorem
K{qr) =Kr.Kq,
i.e. the conjugate of tM product of two quaternions is the product of
their conjugates in inverted order.
56.] The propositions just proved are, of course, true of quater-
nions as well as of versors ; for the former involve only an additional
30 QUATERNIONS. [57-
numerical factor which has reference to the length merely, and not
the direction, of a vector (§48).
57.] Seeing thus that the commutative law does not in general
hold in the multiplication of quaternions, let us enquire whether
the Associative Law holds. That is, itj), q, r be three quaternions,
have we jtj.g'r = j)q.r ?
This is, of course, obviously true if jo, q, r be numerical quantities,
or even any of the imaginaries of algebra. But it cannot be con-
sidered as a truism for symbols which do not in general give
M = iP-
58.] In the first place we remark that ^, q, and r may be con-
sidered as versors only, and therefore represented by arcs of great
. circles, for their tensors may obviously (§ 48) be divided out from
both sides, being commutative with the versors.
Let AB =p, BB = CA = q, aadi IE =z r.
Join BC and produce the great circle till it meets EF in H, and
make KH = FE = r, Bxidi EG = GB = pq (§ 54).
Join GK. Then
KG=HG.n]: = pq.r.
Join FB and produce it to
meet AB in M. Make
lM=fb,
and MN=AB,
~^frlf^ and join NL. Then
LN= MN.£M = p.qr.
Hence to shew that p.qr = pq.r
all that is requisite is to prove that LN, and KG, described as
above, are equal a/rcs of the same great circle, since, by the figure,
they are evidently measured in the same direction. This is perhaps
most easily efiected by the help of the fundamental properties of
the curves known as Sjokerical Conies. As they are not usually
familiar to students, we make a slight digression for the purpose of
proving these fundamental properties ; after Chasles, by whom and
Magnus they were discovered. An independent proof of the asso-
ciative principle will presently be indicated, and in Chapter VII
we shall employ quaternions to give an independent proof of the
theorems now to be established.
59.*] Dbf. a spherical conic is the curve of intersection of a cone
of the second degree with a sphere, the vertex of the cone being the
centre of the sphere.
59-J
PRODUCTS AND QUOTIENTS OF VKCTORS.
31
Lemma. If a cone have one silHes of circular sections, it bas
another series, and any two circles belonging to different series lie
on a sphere. This is easily proved as follows.
Describe a sphere, A, cutting the cone in one circular section,
C, and in any other point whatever, and let the side OpP of the
cone meet A in p, P; P being a point in C. Then PO-Op is
constant, and, therefore, since P lies in a plane, p lies on a sphere,
a, passing through 0. Hence the locus, c, of js is a circle, being
the intersection of the two spheres A and a.
Let OqQ be any other side of the cone, q and Q being points in
c, C respectively. Then the quadrilateral qQPp is inscribed in a
circle (that in which its plane cuts the sphere, J) and the exterior
angle at p is equal to the interior angle at Q. If OL, OMhe the
lines in which the plane POQ cuts the cyclic planes (planes through
parallel to the two series of circular sections) they are obviously
parallel to pq, QP, respectively ; and therefore
LLOp = LOpq = loqp = iMoq.
Let any third side,
OrE, of the cone be
drawn, and let the
plane OPR cut the
cyclic planes in 01, Om
respectively. Then,
evidently,
L10L= L qpr,
Z.MOm= LQPR,
and these angles are independent of the position of the points p and
P, if Q and R be fixed points.
In a section of thp above
diagram by a sphere whose
centre is 0, IL, Mm are the
great circles which repre-
sent the cyclic planes, PQ,R
is the spherical conic which
represents the cone. The
point P represents the line OpP, and so with the others. The
propositions above may now be stated thus
Arc PL = arc MQ ;
and, if Q and R be fixed, Mm and IL are constant arcs whatever be
the position of P.
32
QUATERNIONS.
[60.
60.] The application to § 58 is now obvious. In the figure of
that article we have
FE=ES, EI) = CA, Sg = CB, LM = FI).
Hence L, C, G, D are points of a spherical conic whose cyclic
planes are those of AJB, FE. Hence also KG passes through L,
and with LM intercepts on AB an arc equal to AB. That is, it
passes through N, or KG and LN are arcs of the same great circle :
and they are equal, for G and L are points in the spherical conic.
Also, the associative principle holds for any number of quaternion
factors. For, obviously,
qr.st = qrs.t = &c., &c.,
since we may consider qr as a single quaternion, and the above
proof applies directly.
61.] That quaternion addition, and therefore also subtraction,
is commutative, it is easy to shew.
For if the planes of two quaternions,
q and r, intersect in the line OA, we
may take any vector OA in that line,
and at once find two others, OB and
OC, such that
0B= qOA,
and OC=r OA.
And {q + r)0A=0B+0C=0C+0B={r+q)6A,
since vector addition is commutative (§ 27).
Here it is obvious that {q + r)OA, being the diagonal of the
parallelogram on OB, OC, divides the angle between OB and OC
in a ratio depending solely on the ratio of the lengths of these
lines, i. e. on the ratio of the tensors of q and r. This will be useful
to us in the proof of the distributive law, to which we proceed.
62.] Quaternion multi-
JD plication, and therefore di-
vision, is distributive. One
simple proof of this depends
on the possibility, shortly to
be proved, of representing
an^ quaternion as a linear
function of three given rect-
angular unit-vectors. And
when the proposition is thus
established, the associative principle may readily be deduced froin it.
But we may employ for its proof the properties of Spherical
63-] PfiODUCTS AND QUOTIENTS OP VECTORS. 33
Conies already employed in demAsttating the truth of the asso-
ciative principle. For continuity we give an outline of the proof
by this process.
Let BA, GA represent the versors of q and r, and be the great
circle whose plane is that of js.
Then, if we take as operand the vector OA, it is obvious that
U{q + r) will be represented by some such arc as BA where B, B, C
are in one great circle ; for {q + r) OA is in the same plane as q OA
and rOA, and the relative magnitudes of the arcs BB and BC
depend solely on the tensors of q and r. Produce BA, BA, CA to
meet be in b, d, e respectively^ and make
^ M = BA, m= BA, Gc= CA.
Also make b^ = dh = cy =^. Then E, F, G, A lie on a spherical
conic of which BG and be are the cyclic arcs. And, because
bfi = dh = cy, pE, hF, y G, when produced, meet in a point R
which is also on the spherical conic (§ 59*). Let these arcs meet
BG in J, B, K respectively. Then we have
JHz= E^ = pUq,
LH=M =pU{q + r),
KE= Gy =p Ur.
Also fj= BB,
and EL = CB.
And, on comparing the portions of the figure bounded respectively
by HKJ and by AGB we see that (when considered with reference
to their effects as factors multiplying OH and OA respectively)
J) U(^qjf.r) bears the same relation to jo Uq and jo Ur
that lf{q + r)\)G2iXsto Uq&xA Ur.
But T{q + r)U{q + r) = q + r = TqUq + TrUr.
Hence T^ + r).jpU{q + r) = Tq.p Uq + Tr.p Ur ;
or, since the tensors are mere numbers and commutative with all
other factors, p{q + r) = pq +pr.
In a similar manner it may be proved that
{q + r)p = qp + rp.
And then it follows at once that
(p + q) (r+s) =pr +ps + qr-j-qs.
63.1 By similar processes to those of § 53 we see that versors,
and therefore also quaternions, are subject to the index-law
q'^.q" = j'"+",
at least so long as m and n are positive integers.
D
34 QUATERNIONS. [64.
The extension of this property to negative and fractional ex-
ponents must be deferred until we have defined a negative or
fractional power of a quaternion.
64.] We now proceed to the special case of guadrantal versors,
from whose properties it is easy to deduce all the foregoing results
of this chapter. These properties were indeed those whose in-
vention by Hamilton in 1843 led almost intuitively to the esta-
blishment of the Quaternion Calculus. We shall content ourselves
at present with an assumption, which will be shewn to lead to
consistent results-; but at the end of the chapter we shall shew
that no other assumption is possible, following for this purpose a
very curious quasi-metaphysical speculation of Hamilton.
65.] Suppose we have a system of three mutually perpendicular
unit- vectors, drawn from one point, which we may call for short-
ness I, J, K. Suppose also that these are so situated that a positive
(i. e. left-handed) rotation through a right angle about / as an axis
brings J to coincide with K. Then it is obvious that positive
quadrantal rotation about / will make K coincide with /; and,
about K, will make I coincide with /.
For definiteness we may suppose J to be drawn eastwards, J north-
wards, and K upwards. Then it is obvious that a positive (left-
handed) rotation about the eastward line (7) brings the northward
line (i7) into a vertically upward position {K); and so of the others.
66.] Now the operator which turns J into Z" is a quadrantal
versor (§ 53) ; and, as its axis is the vector I, we may call it i.
Thus T'^^' °^ K=iJ- (1)
Similarly we may put -= =j, or I=.jK, (2)
and -Y = k, or J = hi. (3)
[It may here be noticed, merely to shew the symmetry of the
system we are explaining, that if the three mutually perpendicular
vectors /, /, Xbe made to revolve about a line equally inclined to
all, so that / is brought to coincide with J, J will then coincide
with K, and X with I: and the above equations will still hold good,
only (1) will become (2), (2) will become (3), and (3) will become
(I)-]
67.] By the results of § 50 we see that
-/_ K
K ~ J'
69.]
PEODUCTS AND QUOTIENTS OP VEOTOES.
35
1. e. a southward unit- vector bears the same ratio to an upward
unit-vector that the latter does to a northward one; and therefore
we have
~" -J=iK.
— K
Similarly -^^ =
and
-/
K
-E
-I
J
t, or
or
= ff, or
-K=jl;
-I=kJ.
■(4)
.(5)
(6)
(7)
68.] By (4) and (1) we have.
-J =iK= i{iJ) = i^J.
Hence p = _ 1
And, in the same way, (5) and (2) give
-^'=-1' (8)
and (6) and (3) F ^ — 1 (9^
Thus, as the directions of /, J, K are perfectly arbitrary, we see
that Ue square of every quad/rantal versor is negative unity.
Though the following proof is in principle exactly the same as
the foregoing, it may perhaps be of use to the student, in shewing
him precisely the nature as well as the simplicity of the step we
have taken.
Let ABA' be a semicircle, whose centre
is 0, and let OB be perpendicular to AOA'.
Then ^=:^ , = q suppose, is a quadrantal
OA
OA'
versor, and is evidently equal to -:=r;
§§ 50, 53. _^ _ ^^
OA' OB 61'
A' -ot
Hence
r = -=-
OB
OA OA
69.] Having thus found that the squares of i, J, h are each equal
to negative unity ; it only remains that we find the values of their
products two and two. For, as we shall see, the result is such as
to shew that the value of any other combination whatever of i, j, Jc
(as factors of a product) may be deduced from the values of these
squares and products.
Now it is obvious that
_Z
-I~ K
D a
_I__ .
36 QUATERNIONS. [70.
(i. e. the versor which turns a westward unit- vector into an upward
one will turn the upward into an eastward unit) ;
or K = J{-I)=-jI* (10)
Now let us operate on the two equal vectors in (10) by the same
versor, i, and we have
iK = i {—jl) = —ijl.
But by (4) and (3)
iK = -J =-kI.
Comparing these equations, we have
—ijl=-kl;
or, by § 54 (end), ij = k,^
and symmetry gives jh = i, \ (11)
hi = j. J
The meaning of these important equations is very simple ; and is,
in fact, obvious from our construction in § 54 for the multiplication
of versors; as we see by the annexed figure, where we must re-
member that i, j, ^^re quadrantal versor s whose planes are at right
angles, so that the figure represents a
hemisphere divided into quadrantal tri-
angles.
Thus, to shew that ij = k, we have,
being the centre of the sphere, N, E,
S, W the north, east, south, and west,
and ^the zenith (as in § 65) ;
j6W= 6z,
whence ijOW=^iOZ= OS - kOW.
70.] But, by the same figure,
i_ON=OZ, _ _ _
whence jiON = jOZ= OE = -OW=-kON.
71. J From this it appears that
ji=-k, \
and similarly kj =- — i, > (12)
ik = -J, )
and thus, by comparing (11),
(/ = -i* = ^> )
jk=~kj=iA ((11), (12)).
ki = —ik = J. )
* The negative sign, being a mere numerical &ctor, is evidently commutative with
j ; indeed we may, if necessary, easily assure ourselves of the fact that to turn the
negative (or reverse) of a vector through a right (or indeed any) angle, is the same
thmg ae to turn the vector through that angle and then reverse it.
74-] PRODUCTS ANB QUOTIENTS OF VECTOBS. 37
These equations, along with
i2=/=F=-l ((7), (8), (9)),
contain essentially the whole of Quaternions. But it is easy to see
that, for the first group, we may substitute the single equation
V^=-l, (13)
since from it, by the help of the values of the squares of i, J, h, all
the other expressions may be deduced. We may consider it proved
in this way, or deduce it afresh from the figare above, thus
hON= 6W,
jkON= j6W= 6Z,
ijhON= ijOW=^i6Z= 68 = -0N.
72.] One most important step remains to be made, to wit the
assumption -referred to in § 64. We have treated i,j, k simply as
quadrantal versors ; and /, J, K as unit-vectors at right angles to
each other, and coinciding with the axes of rotation of these versors.
But if we collate and compare the equations just proved We have
= .k, (11)
.iJ= K, (1)
\Ji=-k, (12)
Ijl = -K, (10)
with the other similar groups symmetrically derived from them.
Now the meanings we have assigned to i, _;', k are quite inde-
pendent of, and not inconsistent with, those assigned to I, J, Ki
And it is superfluous to use two sets of characters when one will
suffice. Hence it appears that «', /, k may be substituted for J, /, K;
in other words, a unit-vector when employed as a factor may be con^
sidered as a quadrantal versor whose plane is perpendicular to the
vector. This is one of the main elements of the singular simplicity
of the quaternion calculus.
73.] Thus the product, and therefore the quotient, of two perpen-
dicular vectors is a third vector perpendicular to hoth.
Hence the reciprocal (§ 51) of a vector is a vector which has the
opposite direction to that of the vector, and its length is the re-
ciprocal of the length of the vector.
The conjugate (§ 52) of a vector is simply the vector reversed.
Hence, by § 52, if a be a vector
{Taf = aKa = a (-a) = -a".
74.J We may now see that every versor may be represented by
a power of a unit-vector.
38 QUATEENIONS. [75-
For, if a be any vector perpendicular to i (which is an^ definite
unit-vector),
»a, = /3, is a vector equal in length to a, but perpendicular
to both i and a ;
i^a = — a,
i^a =—ia = — /3,
i*a = — i/3 = —i^ a = a.
Thus, by successive applications of i, a is turned round i as an axis
through successive right angles. Hence it is natural to define i*" as
a versor which turns any vector perpendicular to i through m right
angles in the positive direction of rotation about i as an axis. Here m
may have any real value whatever, whole or fractional, for it is
easily seen that analogy leads us to interpret a negative value of m
as corresponding to rotation in the negative direction.
75.] From this again it follows that any quaternion may be
expressed as a power of a vector. For the tensor and versor elements
of the vector may be so chosen that, when raised to the same power,
the one may be the tensor and the other the versor of the given
quaternion. The vector must be, of course, perpendicular to the
plane of the quaternion.
76.] And we now see, as an immediate result of the last two
sections, that the index-law holds with regard to powers of a
quaternion (§ 63).
77.] So far as we have yet considered it, a quaternion has been
regarded as the product of a tensor and a versor : we are now to
consider it as a sum. The easiest method of so analysing it seems
to be the following.
Let ^=- represent any quaternion. Draw
BC perpendicular to OA, produced if neces-
sary.
Then, §19, OB = OC+CB.
But, § 22, OC = xOA,
where a; is a number, whose sign is the same
as that of the cosine of Z AOB.
Also, § 73, since CB is perpendicular to OA,
CB = yOA,
where y is a vector perpendicular to OA and CB, i.e. to the plane
of the quaternion.
TT OB (vOl + yOA
Hence -^^^ = =i — =a! + v.
OA OA
79-] PRODUCTS AND QUOTIENTS OF VECTORS. 39
Thus a quaternion, in general, mSy be decomposed into the sum of
two parts, one numerical^ the other a vector. Hamilton calls them
the SCALAE, and the vector, and denotes them respectively by the
letters S and T prefixed to the expression for the quaternion.
78.] Hence q = Sq+ Vq, and if in the above example
OB
M=^
then OB = 0C+ CB = Sq.OA-\- Fq.Ol*.
The equation above gives
OC^Sq.OA,
GB = rq.OA.
79.] If, in the figure of last section, we produce BG to B, so as
to double its length, and join OB, we have, by § 52,
^=Kq = SKq^YKq;
.-. 6B=0C + CB = 8Kq.62+rKq.0J.
Hence OC = SKq.OA,
and CB = rKq.OA.
Comparing this value of OC with that in last section, we find
8Kq = 8q, (1)
or the scalar of the conjugate of a quaternion is equal to the scalar of
the quaternion.
Again, CB — — CB by the figure, and the substitution of their
values gives VKq^-Vq, '. (2)
or the vector of the conjugate of a quaternion is the vector of the
quaternion reversed.
We may remark that the results of this section are simple con-
sequences of the fact that the symbols S, V, K are commutative f. ^
Thus SKq = K8q = Sq,
since the conjugate of a number is the number itself; and
VKq = KVq = -rq{\ 73).
* The points are inserted to shew that S and Y apply only to q, and not to qOA .
+ It is curious to compare the properties of these quaternion symbols with those of
the Elective Symbols of Logic, as given in Boole's wonderful treatise on the LoAm of
Thought; and to think that the same grand science of mathematical analysis, by
processes remarkably similar to each other, reveals to ns truths in the science of
position far beyond the powers of the geometer, and truths of deductive reasoning to
which unaided thoug'ht could never have led the logician.
40 QUATERNIONS. [8o.
Again, it is obvious that
:^Sq = S2q, l,Fq= Flq,
and thence SKq = Kl,q.
80.] Since any vector whatever may be represented by
xi + yj-i-zl!
where x, y, z are numbers (or Scalars), and i, j, h may be any three
non-coplanar vectors, §§ 23, 25 — though they are usually under-
stood as representing a rectangular system of unit-vectors — and
since any scalar may be denoted by w; we may write, for any
quaternion q, the expression
q = w-\-m-\-yj->rzh (§ 78).
Here we have the essential dependence on four distinct numbers,
from which the quaternion derives its name, exhibited in the most
simple form.
And now we see at once that an equation such as
where §^= v/+x'i-\-i^j-\-/h,
involves, of course, the ybwr equations
vf=w, af= X, y'=y, i^—z.
81.] "We proceed to indicate another mode of proof of the dis-
tributive law of multiplication.
We have already defined, or assumed (§61), that
- + - = >
a a a
or ^a-i+ya-i = (^ + y)a-i,
and have thus been able to understand what is meant by adding
two quaternions.
But, writing o for a~^, we see that this involves the equality
[fi + y)a = /3a + ya;
from which, by taking the conjugates of both sides, we derive
And a combination of these results (putting /3 + y for a' in the
latter, for instance) gives
(^ + y)(^+/) = (/3+y)^+G3 + y)/
= i3/3'-|- yi3'+ /3y'+ yy by the former.
Hence the distributive principle is true in the multiplication of vectors.
It only remains to shew that it is true as to the scalar and
83.] PEODUCTS AND QUOTIENTS OF VECTOES. 41
vector parts of a quaternion, and then we shall easily attain the
general proof.
Now, if a be any scalar, a any vector, and q^ any quaternion,
(a + aj 3' = «§' + aq.
For, if ;3 be the vector in which the plane of §' is intersected by
a plane perpendicular to a, we can find other two vectors, y and 8,
in these planes such that
And,. of course, a may be written ■—; so that
^ ^^ y3 8 8
-'^6 +6-'^8 + ^ 6
= aq-\-aq^.
And the conjugate may be written
/(a' + o') = ?V + /a' (§55).
Hence, generally,
(a + a)(3 + ;8) =ffli + a/3 + Ja+a;8;
or, breaking up « and 5 each into the sum of two scalars, and a, /3
each into the sum of two vectors,
(«i + aa + oi + Og) ( *! + «2 + ^1 + /Sg)
= K + «2) (*i + ^2) + («i + «.) (/3i + /Sz) + {\ + ^2) («i + "2)
+ («l + «2)(^l + /32)
(by what precedes, all the factors on the right are distributive, so
that we may easily put it in the form)
= («i + «i) (^1 + /5i) + K + «i) (*2 + /32) + («2 + a^) («i + /3i)
+ («2+a2)(52 + /32)-
Putting ai + ai=j9, «2 + a2 = $', ^i + ZSj = /, 5a + /32=»,
we have ( J" + ?) 0" + *) =/'>" + i'»+$'»' + S'*.
83.] For variety, we shall now for a time forsake the geometrical
mode of proof we have hitherto adopted, and deduce some of our
next steps from the analytical expression for a quaternion given
in § 80, and the properties of a rectangular system of unit-vectors
as in § 71.
We will commence by proving the result of § 77 anew.
83.] Let a = xi + yj-\-zh.
42 QUATERNIONS. [84.
Then, because by § 7 1 every product or quotient of i, J, h is reducible
to one of them or to a number, we aye entitled to assume
^ = ^ = a) + ^J + 77y+C^,
where co, f, 77, f are numbers. This is the proposition of § 80.
84.] But it may be interesting to find to, £, tj, f in terms of x, y, z,
af,^, z.
We have ^ = qa,
or x'i-\-y'j-\-^k = (a, +^»4- v'+ C^) {xi + yj+zk)
as we easily see by the expressions for the powers and products of
hji '^j given in § 71. But the student must pay particular attention
to the order of the factors, else he is certain to make mistakes.
This (§ 80) resolves itself into the four equations
0= ^SB + riy + Cm,
x'=(x,x +i?«— fy,
^=wy—iz +Ca?,
/= a)Z + $y—r]X.
The three last equations give
xx'+yy' + zz'= a) {x^ + y^ + z^),
which determines m.
Also we have, from the same three, by the help of the first,
ix' + riy'+C/=0;
which, combined with the first, gives
^ = ■n ^ C .
y/—zy' zuZ—x/ x^—yx^'
and the common value of these three fractions is then easily seen
to be 1
x'^+y^+ z^
It is easy enough to interpret these expressions by means of
ordinary coordinate geometry : but a much simpler process will
be furnished by quaternions themselves iti the next chapter, and, in
giving it, we shall refer back to this section.
85.] The associative law of multiplication is now to be proved
by means of the distributive (§ 81). We leave the proof to the
student. He has merely to multiply together the factors
w + xi + yj + zk, w'+x'i + yj+Zk, and w" + x"i + y"j+/'k,
as follows : —
First, multiply the third factor by the second, and then multiply
the product by the first ; next, multiply the second factor by the
88.J PRODUCTS AND QUOTIENTS OP VECTORS. 43
first and employ the product to multiply the third : always re-
membering that the multiplier in any product is placed lefore the
multiplicand. He will find the scalar parts and the coefficients of
i,j, Te, in these products, respectively equal, each to each.
86.] With the same expressions for a, )3, as in section 83, we have
a^ = {xi + yj + zh) {x'i + y'j + ^h)
= - [xx' +yy' + zz') + ly/ -zy')i + {zx'-x^)j-\- {xy" -yx')k.
But we have also
^a= — {xx'+ yy' +zz')- {y/ — z/) i - {zaf - xz')j — (a/ - yx) k
The only difference is in the sign of the vector parts.
Hence Sa^ = Spa, (1)
ral3=-r^a, (2)
also afi + ^a = 2Sa^, (3)
al3-^a = 2rap, (4)
and, finally, by § 79, a^ = K^a (5)
87.] If a = /3 we have of course (§ 25)
x = x', y=y', z = z',
and the formulae of last section become
a^ — ^a = a' = —(x' + y^ + z') ;
vsrhich was anticipated in § 73, where we proved the formula
and alsOj to a certain extent, in § 25.
88.] Now let q and / be any quaternions, then
S.qr = S.{Sq+ Vq) {Sr+ Fr),
= S.{SqSr+Sr.Vq + Sq.Fr+ FqFr),
= SqSr+SFqFr,
since the two middle terms are vectors.
Similarly, S.rq = SrSq + SFr Fq.
Hence, since by (1) of § 86 we have
SFqFr = SFrFq,
we see that S.qr = S.rq, (1)
a formula of considerable importance.
It may easily be extended to any number of quaternions, because,
r being arbitrary, we may put for it rs. Thus we have
S.qrs = S.rsq,
= S.sqr
by a second application of the process. In words, we have the
theorem — tAe scalar of ike product of any number of given quaternions
depends only upon the cyclical order in which they are arranged.
44 QUATERNIONS. [89.
89.] An important case is that of three factor.s, each a vector.
The formula then becomes
S.a^y = S.^ya = S.ya^.
But S. aPy = Sa{Spy + V^y)
= SaVfiy, since aSfty is a vector,
= -Saryp, by (2) of §86,
= -Sa{SyP+ryP)
= -S.ayfi.
Hence tke scalar of the product of three vectors changes sign when the
cyclical order is altered.
Other curious propositions connected with this will be given
later, as we wish to devote this chapter to the production of the
fundamental formulae in as compact a form as possible.
90.] By (4) of §86,
2F^y=:/3y-yi8.
Hence 2FaFfiy = Fa {fiy — y^)
(by multiplying both by a, and taking the vector parts of each side)
= F{apy+pay—pay—ayfi)
(by introducing the null term ^ay—^ay).
That is
2rar^y=r.{aP + l3a)y-F(fiSay + l3Fay + Say.p+Fay.p)
= F{2Sa^)y-2F^Say
(if we notice that F.Fay.^ =~ F^Fay, by (2) of § 86).
Hence FaF^y = ySa^—fiSya, (1)
a formula of constant occurrence.
Adding aS^y to both sides we get another most valuable formula
F.afiy = aSfiy-^Sya + ySa^; (2)
and the form of this shews that we may interchange y and o
without altering the right-hand member. This gives
F.a^y = F.y^a,
a formula which may be greatly extended.
91.] We have also
FFa^Fyh = - FFyb Fafi by (2) of § 86 :
= bSyFap-ySbFafi = bS.afiy-yS.a^b,
= - pSa, Fyh + axS/3 Fyh = -pS. ayh + aS. j8y8,
all of these being arrived at by the help of § 90 (1) and of § 89 ;
and by treating alternately Fa/3 and Fyb as simple vectors.
Equating two of these values, we have
bS.al3y = aS.I3yb + l38.yab + yS.al3t, (3)
93-J PRODUCTS AND QUOTIENTS OF VECBOES. 45
a very useful formula, expressing any vector whatever in terms
of three given vectors.
93.] That such an expression is possible we knew already by
§ 23. For variety we may seek another expression of a similar
character^ by a process which differs entirely from that employed
in last section.
a, /3, y being any three vectors, we may derive from them three
others Faj3, V^y, Vya ; and, as these will not generally be coplanar,
any other vector 6 may be expressed as the sum of the three, each
multiplied by some scalar (§ 23). It is required to find this ex-
pression for 5.
Let h=go FajS + i/Vfiy + z Vya.
Then 8yb = xS.yajS = xS.a^y,
the terms in ^ and z going out, because
Syri3y = S.yPy = S/Sy^ = y^SjS = 0,
for y^ is (§ 73) a number.
Similarly /S/38 = zS.^ya = zS.aj3y,
and Sab = yS.a^y.
Thus bS.apy = Fa^Syb + FjSySab + VyaS^b (4)
93.] We conclude the chapter by shewing (as promised in § 64)
that the assumption that the product of two parallel vectors is
a number, and the product of two perpendicular vectors a third
vector perpendicular to both, is not only useful and convenient,
but absolutely inevitable, if our system is to deal indifferently with
all directions in space. We abridge Hamilton's reasoning.
Suppose that there is no direction in space pre-eminent, and
that the product of two vectors is something which has quantity,
so as to vary in amount if the factors are changed, and to have
its sign changed if that of one of them is reversed ; if the vectors
be parallel, their product cannot be, in whole or in part, a vector
inclined to them, for there is nothing to determine the direction
in which it must lie. It cannot be a vector parallel to them ; for
by changing the sign of both factors the product is unchanged,
whereas, as the whole system has been reversed, the product vector
ought to have been reversed. Hence it must be a number. Again,
the product of two perpendicular vectors cannot be' wholly or partly
a number, because on inverting one of them the sign of that
number ought to change; but inverting one of them is simply
equivalent to a rotation through two right angles about the other,
and (from the symmetry of space) ought to leave the number
46 QUATERNIONS.
unchanged. Hence the product of two perpendicular vectors must
be a vector, and a simple extension of the same reasoning shews
that it must be perpendicular to each of the factors. It is easy
to carry this farther, but enough has been said to shew the character
of the reasoning.
EXAMPLES TO CHAPTER II.
1 . It is obvious from the properties of polar triangles that any
mode of representing versors by the sides of a triangle must have
an equivalent statement in which they are represented by angles in
the polar triangle.
Shew directly that the product of two versors represented by
two angles of a spherical triangle is a third versor represented
by the supplement of the remaining angle of the triangle; and
determine the rule which connects the directions in which these
angles are to be measured.
2. Hence derive another proof that we have not generally
m = IP-
3. Hence shew that the proof of the associative principle, § 57,
may be made to depend upon the fact that if from any point of
the sphere tangent arcs be drawn to a spherical conic, and also arcs
to the foci, the inclination of either tangent arc to one of the focal
arcs is equal to that of the other tangent arc to the other focal arc.
4. Prove the formulae
2S.apy = a^y—y^a,
2r.a/3y= a^y + y^a.
5. Shew that, whatever odd number of vectors be represented by
a, 13, y, &c., we have always
F.aPybe = V.eby^a,
KajaybeCn = r.7jfe8y/3a, &c.
6. Shew that
S.FajSFfiY^ya = -{S-ajSyf,
r. ral3 r^y Vya = TajS {y^Sa^ - SfiySya) + ,
and F. ( Fa^ F. Ffiy Fya) = (fiSay- aSfiy) S.a^y.
7. If a, /3, y be any vectors at right angles to each other, shew that
(a3 + ^3 + y 3) ;S.a^y = a* Fj3y + /3* Fya + y* Fafi.
EXAMPLES TO CHAPTER 11. 47
8. If a, j3, y be non-eoplanar veifcors, find the relations among
the six scalars, x, y, z and f, t\, f, which are implied in the equation
xa->ry^-^zy = iFfSy + r\ Fya + QFa^.
9. If a, j3, y he any three non-eoplanar vectors, express any
fourth vector, 6, as a linear function of each of the following sets of
three derived vectors,
r.yap, V.afiy, V.^ya,
and V.ra^r^yVya, FV^yVyaVa^, F.FyaFapFfiy.
10. Eliminate p from the equations
Sap = a, Sj3p = b, Syp = c, Sbp = d,
where a, /3, y, 6 are vectors, and a, b, c, d scalars.
11. In any quadrilateral, plane or gauche, the sum of the squares
of the- diagonals is double the sum of the squares of the lines joining
the middle points of opposite sides.
CHAPTER IIL
INTERPKETATIONS AND TRANSFORMATIONS OF
QUATERNION EXPRESSIONS.
94.] Among the most useful characteristics of the Calculus of
Quaternions, the ease of interpreting its formulae geometrically,
and the extraordinary variety of transformations of which the
simplest expressions are susceptible, deserve a prominent place.
We devote this Chapter to some of the more simple of these, to-
gether with a few of somewhat more complex character but of
constant occurrence in geometrical and physical investigations.
Others will appear in every succeeding Chapter. It is here,
perhaps, that the student is likely to feel most strongly the peculiar
difficulties of the new Calculus. But on that very account he
should endeavour to master them, for the variety of forms which
any one formula may assume, though puzzling to the beginner, is
of the most extraordinary advantage to the advanced student, not
alone as aiding him in the solution of complex questions, but as
affording an invaluable mental discipline.
95.] If we refer agiain to the figure of § 77 we see that
0C= OB cos AOB,
CJB = OB sin JOB.
Hence, if OJ = a, OB = p, and /.AOB = 6, we have
OB = Tl3, OA = Ta,
OC = Tfi cos d, CB = Tfi sin 0.
o/3 OC Tl3
Hence S- = -^-r = -y==— cos5.
a OA la
Similarly rr^ = ^ = ^sin..
9 7- J INTERPRETATIONS AND TRANSFORMATIONS. 49
Hence, if e be a unit- vector perpenoicular to o and /3, or
UOA o.
we have F- = -^ sin 0.{.
a Ta
96.] In the same way we may shew that
TVa^ = Ta Tfi sin 6,
and Fa^ = Ta Tfi sin O.r,
where 77= Urap = UF^-
a
Thus tAe scalar of the product of two vectors is the continued product
of their tensors and of the cosine of the sitpplement of the contained
angle.
The tensor of the vector of the product of two vectors is the con-
tinued product of their tensors and the sine of the contained angle ;
and the versor of the same is a unit-vector perpendicular to both, and
such that the rotation about it from the first vector (i. e. the multiplier')
to the second is left-handed or positive.
Hence TFa^ is doMe the area of the triangle two of whose sides
are a, /3.
97.]
(a.) In any triangle A£C we have
AC = A£ + W.
Hence IC^ = SAC AC = S.AC{AB + SC).
With the usual notation for a plane triangle the interpretation
of this formxila is
—b^ = —be cos A— ab cos, C,
or b= a cos C+c cos A
(b.) Again we have, obviously,
rABAC= FAS{A£ + BC)
= FABBG,
or cb sin A = ca sin B,
sin A sin B sin C
whence = — j— = •
a c
These are truths, but not truisms, as we might have been led
to fancy from the excessive simplicity of the process employed.
E
50 QUATERNIONS. [98.
98.] From § 96 it follows that, if a and /3 be both actual (i. e.
real and non-evanescent) vectors, the equation
Sa^ =
shews that cos 6 = 0, or that a is perpendicular to /3. And, in fact,
we know already that the product of two perpendicular vectors is a
vector.
Again, if ^„^ = 0,
we must have sin ^ = 0, or a is parallel to /3. We know already
that the product of two parallel vectors is a scalar.
Hence we see that
Sa^ =
is equivalent to o = Fy/3,
where y is an undetermined vector ; and that
is equivalent to a = a;/3,
where aj is an undetermined scalar.
99.] If we write, as in § 83,
o — ix +ji/ + kz,
/3 = ix'+j/ + M,
we have, at once, by § 86,
Sa^ = —xaf—yy'—z/
, ^x af , y 1/ z z' \
= -r/( - + I-£^ + — )
\ r r r r r r '
where r = -s/as^+j^^ +«^ /= \/«'^+y^+/^.
Also r^ = ^|i:!:^i+ff:^-y+at.er,j.
These express in Cartesian coordinates the propositions we have
just proved. In commencing the subject it may perhaps assist
the student to see these more familiar forms for the quaternion
expressions ; and he will doubtless be induced by their appearance
to prosecute the subject, since he cannot fail even at this stage to
see how much more simple the quaternion expressions are than
those to which he has been accustomed.
100.] The expression S.a&y
may be written S ( Fa/3) y,
because the quaternion a.^y may be broken up into
of which the first term is a vector.
102.] INTERPEETATIONS AND TEANSFOEMATIONS. 51
But, by §96,
S ( ra/3) y = TaTl3 sin 9 Sriy.
Here Tr) = 1, let (|) be the angle between ?j and y, then finally
S.a^y =-TaT^Ty sin 5 cos (|).
But as ?j is perpendicular to a and /3, Ty cos ^ is the length of the
perpendicular from the extremity of y upon the plane of a, /3. And
as the product of the other three factors is (§ 96) the area of the
parallelogram two of whose sides are a, ^, we see that the mag-
nitude otS.apy, independent of its sign, is i^e volume of the parallel-
epiped of which three coordinate edges are a, fi, y; or six times the
volume of the pyramid which has a, ^, y for edges.
101.] Hence the equation
S.apy = 0,
if we suppose a, /S, y to be actual vectors, shews either that
sin e = 0,
or cos(^ = 0,
i. e. two of the three vectors are parallel, or all three are pArallel to
one plane.
This is consistent with previous results, for if y = ^j3 we have
S.aPy=:pS.afi^ = Q;
and, if y be coplanar with a, fi, we have y =pa + qP, and
S.al3y = S.al3{pa + ql3) = 0.
102.] This property of the expression S.a^y prepares us to find
that it is a determinant. And, in fact, if we take a, ;3 as in § 83,
and in addition ^ ^ ^^" +_^y ^ ^/'^
we have at once
S.apy = —x" [yi^-zy')-f {zx'—xz) ^z" {x/ —yx'),
=.— X y z
x' y' /
of' f z"
The deterrhinant changes sign if we make any two rows change
places. This is the proposition we met with before (§ 89) in the
form s^afiy = ^S.jSay = S.^ya, &e.
If we take three new vectors
ai = ix+j'a^+^a/',
yi = iz+J/+M',
we thus see that they are coplanar if o, ;3, y are so. That is, if
iS.al3y = 0,
then (S.Oj/Sjyi = 0.
E -2
52 QUATERNIONS. [103.
103.] We have, by § 52,
{Tqf = qKq = {Sq+ Fq) (Sq- fq) (§ 79),
= lSqf-{rqf by algebra,
= {SqY+{Trqf (§73).
liq = aj8, we have Kq = fia, and the formula becomes
a/3.;8a = a''^^ = {Sa^f-{Va^f.
In Cartesian coordinates this is
(a!''+/+02)(a^2+/2+/2)
More generally we have
(r(gr))2 = qrK{qr)
= qrKrKq (§ 55) = {Tqf {Trf (§ 52).
If we write q =.w ■\-a = w +ix +jy + kz,
r = w' + l3 =. w'+iaf+Jy'+k/;
this becomes
= {wio'—xx'—^^'—z/f + {loixf + «/«; +^/—z/)'^
+ {w/ + 'u/y+zx'—x/y + {10/ +w'z+x/—ya/)^,
a formula of algebra due to Euler.
104.] We have, of course, by multiplication,
(a+/3)2 = a^ + aj3 + ^a + P^ = a' + 2Sa^ + fi'' (§86 (3)),
Translating into the usual notation of plane trigonometry, this
becomes c^ =za^-2ah cos C+ b%
the common formula.
Again, r(a+/3) (a-/3) = - rai3+ T/Sa = -2 FaiS (§ 86 (2)).
Taking tensors of both sides we have the theorem, the jparallelogram
whose sides are parallel and equal to the diagonals of a given paral-
lelogram, has double its area (§ 96).
Also iS(a + /3)(a-/3) = a^-^^
and vanishes only when a^ = /3^, or Ta—T^\ that is, the diagonals
of a parallelogram are at right angles to one another, when, and only
when, it is a rhombus.
Later it will be shewn that this contains a proof that the angle in
a semicircle is a right angle.
105.] The expression p = a^a'^
obviously denotes a vector whose tensor is equal to that of /3.
But we have S.^ap = 0,
so that p is in the plane of o, ^.
Also we have Sap = Sa^,
I06.] INTERPRETATIONS AND TRANSFOEMATIONS. 53
•
so that /3 and p make equal angles with a, evidently on opposite
sides of it. Thus if a be the perpendicular to a reflecting surface
and /3 the path of an incident ray, p will be the path of the re-
flected ray.
Another mode of obtaining these results is to expand the above
expression, thus, § 90 (2),
p = 2a-^Sa^-^^
SO that in the figure of § 77 we see that if OA = a, and OJB = ^, we
have OJ) = p = a^a~''^.
Or, agaiuj we may get the result at once by transforming the
equation to U- = U--
106.] For any three coplanar vectors the expression
p = afiy
is (§ 101) a vector. It is interesting to determine what this vector
is. The reader will easily see that if a circle be described about
the triangle, two of whose sides are (in order) a and /3, and if from
the extremity of /3 a line parallel to y be drawn again cutting the
circle, the vector joining the point of intersection with the origin
of a is the direction of the vector afiy. For we may write it in the
form a
p = a^^fi-^y = -{T^fafi-^y = -{T^f -y,
which shews that the versor (-A which turns j3 into a direction
parallel to a, turns y into a direction parallel to p. And this ex-
presses the long-known property of opposite angles of a quadri-
lateral inscribed in a circle.
Hence if a, ^, y be the sides of a triangle taken in order, the
tangents to the circumscribing circle at the angles of the triangle
are parallel respectively to
a^y, Pya, and ya)3.
Suppose two of these to be parallel, i. e. let
a/3y = x^ya = as ay 13 (§ 90),
since the expression is a vector. Hence
Py = xyp,
which requires either
x=\, Fy^ = or y || /3,
a case not contemplated in the problem ;
or a; = -l, S^y = 0,
54 QUATERNIONS, [107.
i. e. the triangle is right-angled. And geometry shews us at once
that this is correct.
Again, if the triangle be isosceles, the tangent at the vertex is
parallel to the base. Here we have
wfi = a^y,
or (X!{a + y) = a{a + y)y;
whence x = y'' = a?, or Ty = Ta, as required.
As an elegant extension of this proposition the reader may prove
that the vector of the continued product a^yS of the vector-sides of
a quadrilateral inscribed in a sphere is parallel to the radius drawn
to the corner {a, 8).
107.] To exemplify the variety of possible transformations even
of simple expressions, we will take two cases which are of frequent
occurrence in applications to geometry.
Thus T{p-\-a) = T{s>-a),
[which expresses that if
02 = a, 0A'= —a, and OP = p,
we have AP = A'F,
and thus that P is any point equidistant from two fixed points,]
may be written (p + a)^ = {p—af,
or p'^ + iSap + a^ = p^ — ^Sap + a^ {^101),
whence Sap = 0.
This may be changed to
ap+ pa = 0,
or ap + Kap = 0,
SU^ = 0,
a
or finally, TFU^ = 1,
all of which express properties of a plane.
Again, Tp = Ta
may be written T - = 1,
^ a'' ^ a''
(p + aY-28a{p + a) = 0,
p= {p + a)-'^a{p+a),
S{p + a){p—a) = 0, or finally,
T.{p + a){p-a) = 2TVap.
I09-] INTERPRETATIONS AND TRANSFORMATIONS. 55
•
All of these express properties of a sphere. They will be in-
terpreted when we come to geometrical applications.
108.] "We have seen in § 95 that a quaternion may be divided
into its scalar and vector parts as follows : —
a a a Ta
where 9 is the angle between the directions of a and /3, and e= UF-
a
is the unit- vector perpendicular to the plane of a and /3 so situated
that positive (i. e. left-handed) rotation about it turns a towards /3.
Similarly we have (§ 96)
0/3 = Sa^ + Fa^
= TaT^{-cose + esin0),
6 and e having the same signification as before.
109.] Hence, considering the versor parts alone, we have
U- = cos6 + t sin d.
a
Similarly U^ = cos (j) + e sincj} ;
(j) being the positive angle between the directions of y and /3, and e
the same vector as before, if a, /3, y be coplanar.
Also we have
U- = cos {d + <t)) + e sin {6 + <(>).
But we have always
-•- = -, and therefore
|3 a a
pa a
or cos (<^ + 5) + e sin ((/)-}- 5) = (cos ^ -f e sin ^) (cos 5 + e sin 0)
= cos (\) cos 5— sin (^ sin 9 + e (sin (pcos6 + cos ^ sin 6),
from which we have at once the fundamental formulae for the
cosine and sine of the sum of two arcs, by equating separately the
scalar and vector parts of these quaternions.
And we see, as an immediate consequence of the expressions
abovcj that
cos me + esmme = (cos -f e sin BJ"
if m be a positive whole number. For the left-hand side is a versor
which turns through the angle m5 at once, while the right-hand
56 QUATERNIONS. [lIO.
side is a versor which effects the same object by m successive turn-
ings each through an angle Q. See § 8.
110.] To extend this proposition to fractional indices we have
only to write - for Q, when we obtain the results as in ordinary
trigonometry.
From De Moivre's Theorem, thus proved, we may of course
deduce the rest of Analytical Trigonometry. And as we have
already deduced, as interpretations of self-evident quaternion trans-
formations (§§97, 104), the fundamental formulae for the solution
• of plane triangles, we will now pass to the consideration of spherical
trigonometry, a subject specially adapted for treatment by qua-
ternions ; but to which we cannot afford more than a very few
sections. (More on this subject will be found in Chap. X, in con-
nexion with the Kinematics of rotation.) The reader is referred to
Hamilton's works for the treatment of this subject by quaternion
exponentials.'
111. J Let a, /3, y be unit-vectors drawn from the centre to the
corners A, JB,C oi a triangle on the unit-sphere. Then it is evident
that, with the usual notation, we have (§ 96),
Sa^ = — cos c, Sfiy = —cos a, Sya = —cos &,
Trap= sine, TF^y = sin«, TFya= sin 3.
Also UVafi, UFj3y, UFya are evidently the vectors of the corners
of the polar triangle.
Hence S. UFa^ UF^y = cos £, &c.,
TF.UFa^UF^y = BinB, &c.
Now (§ 90 (1)) we have
SFapFpy = S.aF.^Fpy
=:-Sal38fiy + ^^Say.
Remembering that we have
SFa^F^y = TFa^TF^yS.UFapUF^y,
we see that the formula just written is equivalent to
sin a sin c cos B — ■— cos a cos c + cos h,
or cos h = cos a cos c + sin a sin o cos B.
112.] Again, F.Fa^F^y = -fiSa^y,
which gives
TF. FapF^y = S.apy = S.aFfiy = S.^Fya = S.yFa^,
or sin a sin csinB = sin a sin^„ = sin b sin p^ = sin c sinjO„ ;
where ^„ is the arc drawn from A perpendicular to BC, &c.
113.] INTERPRETATIOKS AND TIUNSFOEMATIONS. 57
Hence sin jo„ = sin e sin £,
sin a sin c . _
sm Ml = -. — 5 — sm />,
smo
sin^o = sin a sin S.
113.] Combining the results of the last two sections, we have
Va^.V^y = sin a sin c cos 5— ^ sin a sine sin 5
= sina sine (cos^— /3 sin 5).
Hence U. Va^ V^y — (cos 5—^3 sin B), 1
and U. Fy^r^a = (cos ^+ i3 sin B). )
These are therefore versors which turn the system negatively or
positively about 0£ through the angle £.
As another instance, we have
sin 5
tan^ =
cos 5
_ Tr.Va^r^y
~ S.Va^r^y
_ r.ra^rfiy
'^ s.ra^r^y
Say + SafiSfiy
The interpretation of each of these forms gives a different theorem
in spherical trigonometry.
Again, let us square the equal quantities
F. ajSy and cuS^y— jSSay + ySa^,
supposing a, jS, y to be any unit- vectors whatever. We have
-{KajSyY = S^^y + S^ya + S^afi+2SfiySyaSafi.
But the left-hand member may be written as
T\al3y-S^.a^y,
whence
1-S^.a^y = S^fiy + S^ya + S^afi + 2S^ySyaSa^, ■
or 1 — cos^fl! — cos^S — cos^c + 2 cos a cos i cos c
= sin^a sin^jo„ = &c.
^ sin^asin^3sin^C= &c.,
all of which are well-known formulae.
Such results may be multiplied indefinitely by any one who has
mastered the elements of quaternions.
58
QUATERNIOlirS.
[114.
114.] A curious proposition, due to Hamilton, gives us a qua-
ternion expression for the spherical excess in any triangle. The
following proof, which is very nearly the same as one of his, though
by no means the simplest that can be given, is chosen here because
it incidentally gives a good deal of other information. We leave
the quaternion proof as an exercise.
Let the unit- vectors drawn from the centre of the sphere to
A, B, C, respectively, be a, p, y. It is required to express, as an
arc and as an angle on the sphere, the quaternion
The figure represents an orthographic projection made on a plane
perpendicular to y. Hence G is the centre of the circle BEe. Let
the great circle through A, B meet BBe in E, e, and let BB be a
quadrant. Thus 2?^ represents y (§ 72). Also make BF=AB=pa~\
Then, evidently, ^ ^ ^a-^y,
which gives the arcual representation required.
Let BF cut Be in G. Make Ga = EG, and join B, a, and a, F.
Obviously, as B is the pole of Ee, Ba is a quadrant ; and since
EG — Ca, Ga = EG, a quadrant also. Hence a is the pole oi BG,
and therefore the quaternion may be represented by the angle BaF.
Make C6 = Ga, and draw the arcs P«/3, Pba from P, the pole of
AB. Comparing the triangles Eba and ea(3, we see that Ea = e/3.
But, since P is the pole of AB, F^a is a right angle : and therefore
as i''a is a quadrant, so is F^. Thus AB is the complement of Ba.
or ySe, and therefore „o _ lAB.
1 1 5. J INTEEPEETATIONS AND TRANSFOEMATIONS. 59
Join bA and produce it to c so tnat Ac = hA; join e, P, cutting
AS in 0. Also join c, £, and £, a.
Since Pis the pole of AS, the angles at o are right angles ; and
therefore, by the equal triangles 6aA, go A, we have
aA = Ao.
But a^ = 2AB,
whence oB = B^,
and therefore the triangles coB and Bafi are equal, and c, ^, a lie
on the same great circle.
Produce cA and cB to meet in M (on the opposite side of the
sphere). H and c are diametrically opposite, and therefore cP,
produced, passes through H.
Now Pa = Pb = PH, for they differ from quadrants by the equal
arcs fl/3, ba, oc. Hence these arcs divide the triangle Eab into three
isosceles triangles.
But IPHb + IPHa = LaHb = Ibca.
Also /.Pab = TT—Zcab — Z-PaH,
LPba =. LPab = it- Lcba- LPbH.
Adding, iLPab^lis— Leah — Lcba— Lbca
= IT — (spherical excess oi abc).
But, as LFaj3 and LBae are right angles, we have
angle of /3a~V = ^^aJ) = L^ae — LPab
= \ (spherical excess, of abc).
[Numerous singular geometrical theorems, easily proved ab initio
by quaternions, follow from this : e. g. The arc AB, which bisects
two sides of a spherical triangle abc, intersects the base at the
distance of a quadrant from its middle point. All spherical tri-
angles, with a common side, and having their other sides bisected
by the same great circle (i. e. having their vertices in a small circle
parallel to this great circle) have_equal areas, &e., &c.J
115.] Let 0« = a, Ob = /3', Oc = y', and we have
©^ey(jr=^-^^-^^
^^V Vy'.
= Ca.BA
But FQ is the complement of BF. Hence the angle of the
quaternion , a A ^ /S'v I / /v
Kj') yz') \7)
60 QUATERNIONS. [ll6.
is half the spherical excess of the triangle whose angular points are at
the extremities of the unit-vectors a', ^', y' .
[In seeking a purely quaternion proof of the preceding proposi-
tions, the student may commence by shewing that for any three
unit- vectors we have a.,„
The angle of the first of these quaternions can be easily assigned ;
and the equation shews how to find that of /Sa-^y. But a stUl
simpler method of proof is easily derived from the composition of
rotations.]
116.] A scalar equation in p, the vector of an undetermined
point, is generally the equation of a surface; since we may sub-
stitute for p the expression . _ ^^j
where x is an unknown scalar, and a any assumed unit-vector.
The result is an equation to determine x. Thus one or more points
are found on the vector xa whose coordinates satisfy the equation j
and the locus is a surface whose degree is determined by that of the
equation which gpives the values of x.
But a vector equation in p, as we have seen, generally leads to
three scalar equations, from which the three rectangular or other
components of the sought vector are to be derived. Such a vector
equation, then, usually belongs to a definite number oi points in
space. But in certain cases these may form a line, and even a
surface, the vector equation losing as it were one or two of the
three scalar equations to which it is usually equivalent.
Thus while the equation ap — &
gives at once p _ „-i^^
which is the vector of a definite point (since we have evidently
/Sa/3 = 0) ;
the closely allied equation y^^ _ a
is easily seen to involve g^o _ q
and to be satisfied by p — oT'^R+xa
whatever be x. Hence the vector of any point whatever in the line
drawn parallel to a from the extremity of a~^/3 satisfies the given
equation.
117.] Again, Fap .Fp^ = {FafiY
is equivalent to but two scalar equations. For it shews that Fap
119.] INTERPRETATIONS AND TRANSFORMATIONS, 61
and F)3p are parallel, i. e. p lies in fhe same plane as a and (3, and
can therefore be written (§ 24)
p = asa+^A
where x and _y are scalars as yet undetermined.
We have now Fap = yVafi,
which, by the given equation, lead to
xy =■ \, or y = -, or finally
p = xa+~j3i
w
which (§ 40) is the equation of a hyperbola whose asymptotes are
in the directions of a and ^8.
118.] Again, the equation
r.raprap = o,
though apparently equivalent to three scalar equations, is really
equivalent to one only. In fact we see by § 91 that it may be
written -aS.a^p = 0,
whence, if a be not zero, we have
S.ajSp = 0,
and thus (§101) the only condition is that p is coplanar with a, j3.
Hence the equation represents the plane in which o and )3 lie.
119.] Some very curious results are obtained when we extend
these processes of interpretation to functions of a quaternion,
q = w+p
instead of functions of a mere vector p.
A scalar equation containing such a qtiaternion, along with
quaternion constants, gives, as in last section, the equation of a
surface, if we assign a definite value to w. Hence for successive
values of w, we have successive surfaces belonging to a system;
and thus when w is indeterminate the equation represents not a
surface, as before, but a volume, in the sense that the vector of any
point within that volume satisfies the equation.
Thus the equation {Tqf = a^,
or w'^—p^ = a^,
or ' {Tpf = a^-w^,
represents, for any assigned value of w, not greater than a, a sphere
whose radius is ^/a^ — w^. Hence the equation is satisfied by the
62 QUATEENIONS. [l20.
vector of any point whatever in the volume of a sphere of radius a,
whose centre is origin.
Again, by the same kind of investigation,
where q = w + p, is easily seen to represent the volume of a sphere
of radius* a: described about the extremity of ^ as centre.
Also S{^)-= —a? is the equation of infinite space less the space
contained in a sphere of radius a about the origin.
Similar consequences as to the interpretation of vector equations
in quaternions may be readily deduced by the reader.
120.] The following transformation is enuntiated without proof
by Hamilton {Lectures, p. 587, and Elements, p. 299).
»--i(rY)*5-i = U{rq-\-KrKq).
To prove it, let r~\r^g^)^g~^ = t, then
Tt = 1, and therefore
But {r^ff = rti,
or r'^q^ = rtqrtq^,
or rq^ = tgrt.
Hence KqKr - t-'^KrKqr\
or KrKq = tKqKH.
Thus we have jji^^^ + ^^^^j = tU{qr±KqKr) t,
or, if we put * = U{qr + KqKr),
Ks= ± Ut.
Hence sKs = {Tsf = 1 = ± stst,
which, if we take the positive sign, requires
st= ±\,
or t= +«-!= ±UKs,
which is the required transformation.
[It is to be noticed that there are other results which might
have been arrived at by using the negative sign above ; some in-
volving an arbitrary unit- vector, others involving the imaginary of
ordinary algebra.J
121.] As a final example, we take a transformation of Hamil-
ton's, of great importance in the theory of surfaces of the second
order.
121.] INTERPRETATIONS AND TRANSFORMATIONS. 63
Transform the expression •
in which a, 13, y are any three mutually rectangular vectors, into
the form mt , \ 2
MW + PkV ^
which involves only two vector-constants, t, k.
{T{ip + pK)}^ = {tp+pK){pi + Kp) (§§ 52, 55)
= (l^ 4- K2)p2 + (tpKp +p/Cpt)
= {l^+K^)p^+2S.lpKp
= {l-K)Y+4:SipSKp.
Hence (Sapf + iS^pf+iSypy^^^.p^ + i-^^'P^'P -
(^2-12)2'' ' ^(^2-12)2
But a-2(5ap)2 + y3-2(<S'j3/))2 + y-2(;S'yp)2 = p2 (§§ 25, 73).
Multiply by jB^ and subtract, we get
The left side breaks up into two real factors if ^2 be intermediate
in value to a^ and y^ : and that the right side may do so the term
in p2 must vanish. This condition gives
ft— k)2
fl2 = A^ L^ ■ and the identity becomes
^(aV(l-5) + yV(^-l))p^(aV(l-5)^-yV(^-l)> = 4^.
Hence we must have
lL^^=^(a^il-.^) + yV{^-l)},
where ^ is an undetermined scalar.
To determine j9, substitute in the expression for p^, and we find
= {P^ + -^)(a^-7')-2(tt^+y^) t4|32.
64 QUATERNIONS. [l22.
Thus the transformation succeeds if
1 2(a'= + /)
1 / o?
which gives jo+ - = + 2^/ 2_^ '
1 v^
Hence J^^ = (jj-y) (a^-/) = ± 4^/^^
or (/c2-i2)-i= iTayy.
. . Ta^Ty 1 2'a-yy
^^^"^' ^ = 77=^' 'P=W^^'
and therefore
ro-I-y , , ^2_a2 y2_^2
Thus we have proved the possibility of the transformation, and
determined the transforming vectors i, k.
123.] By diflFerentiating the equation
we obtain, as will be seen in Chapter IV, the following,
where p also may be any vector whatever.
This is another very important formula of transformation ; and
it will be a good exercise for the student to prove its truth by
processes analogous to those in last section. We may merely
observe, what indeed is obvious, that by putting p'= p it becomes
the formula of last section. And we see that we may write, with
the recent values of i and k in terms of a, /3, y, the identity
aSap + l3S^p + ySyp = ^ \J_,2yL
_ (t — Kfp + 2 {iSkp + kSip)
123.] In various quaternion investigations, especially in such
as involve imaginary intersections of curves and surfaces, the old
imaginarj' of algebra of course appears. But it is to be particularly
124-] INTERPRETATIOlirS AND TEANSFOEMATIONS. 65
noticed that this expression is analogous to a scalar and not to a
veetorj and that like real scalars it is commutative in multiplica-
tion with all other factors. Thus it appears, by the same proof as
in algebra, that any quaternion expression which contains this
imaginary can always be broken up into the sum of two parts, one
realj the other multiplied by the first power of v— 1. Such an
expression, viz. ? = /+ V^?",
where ({ and /' are real quaternions, is called a biquaternion.
Some little care is requisite in the management of these expressions,
but there is no new diflBeulty. The points to be observed are : first,
that any biquaternion can be divided into a real and an imaginary
part, the latter being the product of \/— 1 by a real quaternion ;
secondj that this \/ — 1 is commutative with all other quantities in
multiplication ; tbirdj that if two biquaternions be equal, as
we have, as in algebra, /= /, j"= /'j
so that an equation between biquaternions involves in general eight
equations between scalars. Compare § 80.
124.] We have, obviously, since ^/— i is a scalar,
Hence (§103)
= {8q'+^^-i.Sf+ ?Y+ ^/irTr/')('S/+ V^^/'- rq'- sf- 1 Yf)
= (Sq'+ ,y^I\Sff-{rq'+ ^/^^/')^
= {Tq'f - {Tff + 2 aA^aS. ^Kf.
The only remark which need be made on such formulae is this, that
the tensor of a hiquaternion may vanish while both of the component
quaternions are finite.
Thus, if ^/= Tq",
and S.q'Kq"= 0,
the above formula gives
The condition S.^Kq"=
may be written
Kq"=q'-^a, or q"= -aKq'-^=- ^^r
where a is any vector whatever.
6 6 QUATERNIONS. [ 1 2 5 .
Henee Tq' = Tq" = TKq" = ^ ,
and therefore
Tq\Uq'- </::::\Ua.U^) = (l - ^/^^Ua)^
is the general form of a biquaternion whose tensor is zero.
125.] More generally we have, q, r, ^, / being any four real and
non-evanescent quaternions,
{qJr '/^cf) (r+ ^/^T/) = qr-c['/+ ^^Ix^q/ Jf^r).
That this product may vanish we must have
qr = q'/,-
and q/= —q'r.
Eliminating / w;e have qq'~^qr = — /?',
which gives {l'~^s)^ = ~^>
i.e. q = ({a
where a is some unit-vector.
And the two equations now agree in giving
— r = a/,
so that we have the biquaternion factors in the form
/(a+V^) and — (a-^/^)/;
and their product is
-/(a+ ^T-i) (a- sT^y,
which, of course, vanishes.
[A somewhat simpler investigation of the same proposition may
be obtained by writing the biquaternions as
g^C^-^^+y^) and (?-/-i+^/3i)/,
or g'(/'+V^) and (Z'+v'ZIT)/,
and shewing that
5"= — /'= a, where Ta = 1.]
From this it appears that if the product of two biveciors
p + trV — l and p' + ff'v— 1
is zero, we must have
^-ip = _pV-i = Ua,
where a may be any vector whatever. But this result is still more
easily obtained by means of a direct process.
126.] It may be well to observe here (as we intend to avail our-
selves of them in the succeeding Chapters) that certain abbreviated
127.] INTERPRETATIONS AND TRANSFORMATIONS. 67
forms of expression may be used when they are not liable to confuse,
or lead to error. Thus we may write
T^q for {Tqf,
just as we write ^os^fl for (eos Of,
although the true meanings of these expressions are
T{Ta) and cos (eos 0):
The former is justifiable, as T{Ta) = Ta, and therefore T^d is not
required to signify the second tensor (or tensor of the tensor) of a.
But the trigonometrical usage is quite indefensible.
Similarly we may write
S^q for {Sqf, &c.,
but it may be advisable not to use
Sq^
as the equivalent of either of those just written ; inasmuch as it
might be confounded with the (generally) different quantity
S.q^ or S{q^),
although this is rarely written without the point or the brackets.
137.] The beginner may expect to be a little puzzled with the
aspect of this notation at first ; but, as he learns more of the sub-
ject, he will soon see clearly the distinction between such an ex-
pression as S.FapriSy,
where we may omit at pleasure either the point or the first F with-
out altering the value, and the very different one
Sa^.rpy,
which admits of no such changes, without altering its value.
All these simplifications of notation are, in fact, merely examples
of the transformations of quaternion expressions to which part of
this Chapter has been devoted. Thus, to take 3. very simple ex-
ample, we easily see that
S.Va^r^y = SFapr^y = S.a^FjSy = SaF.^Ffiy = -SaF.{Ffiy)p
= SaF.{Fy^)P = S.aF{yP)^ = S.F{yP)pa = SFy^F^a
= S.y^F^a = &c., &c.
The above group does not nearly exhaust the list of even the simpler
ways of expressing the given quantity. We recommend it to the
careful study of the reader. He will find it advisable, at first, to
use stops and brackets pretty freely ; but will gradually learn to
dispense with those which are not absolutely necessary to prevent
ambiguity.
F 2
68' QUATERNIONS.
EXAMPLES TO CHAPTER III.
1. Investigate, by quaternions, the requisite formulse for changing
from any one set of coordinate axes to another; and derive from
your general result, and also from special investigations, the usual
expressions for the following cases : —
(a.) Rectangular axes turned abbut « through any angle.
(b.) Rectangular axes turned into any new position by rota-
tion about a line equally inclined to the three.
(c. ) Rectangular turned to oblique, one of the new axes lying
in each of the former coordinate planes.
2. If Tp = Ta = T^ = 1, and S.a^p = 0, shew by direct transfor-
mations that ^_ jj^p _ „^ j;r(p _^) ^ + ^in-SalB).
Interpret this theorem geometrically.
3. If Sa^ = 0, Ta=T^=l, shew that
(1 +0™)^ = 2 cos^a^;8 = 2Sa^.a^^.
4. Put in its simplest form the equation
pS. Fa^ r^y Fya = aV. Fya Fafi + 6F. Fafi FjSy + c F. Ffiy Fya ;
and shew that a = S.fiyp, &c.
5. Prove the following theorems, and exhibit them as properties
of determinants : —
{a.) S.[a + ^){fi + y){y^a) = 2S.apy,-
{h.) S.Fa^F^yFya = -(S.a^y)'^,
(c.) S.F(a + l3)i^ + y)F{l3 + y){y + a)F{y + a){a+p) = -4(5.a/3y)^
(d.) S. F( Fafi Ffiy) F( Fj3y Fya) F{ Fya Fa^) = - {S.a^y)\
{e.) S.5€C = — \6{8.afiy)*, where
b = F{Fia+l3){^ + y)F(l3 + y)(y + a)),
t = F{FiP+y){y + a)F(y+a)(a + p)),
{:=F{F(y + a)(^a + l3)F{a + l3)(l3 + y)).
6. Prove the common formula for the product of two determi-
nants of the third order in the form
S.a^yS.a^^iyi^ — Saa^ <S/3aj Sya^
Safi, mi3i Syfi,
Sayi Si3yi Syy^
7. If, in § 102, a, j8, y be three mutually perpendicular vectors,
can anything be predicted as to Oi, jSj, yj ? If a, j3, y be rectangular
unil vectors, what of Oj, p^, y^?
EXAMPLES TO CHAPTER III. 69
8. If aj /3, y, a', 13', y be two sets of rectangular unit-vectors^
shew that Saa'= Syfi'SjSy'-S^fi'Syy', Sec, &c.
9. The lines bisecting pairs of opposite sides of a quadrilateral
are perpendicular to each other when the diagonals of the quadri-
lateral are equal.
10. Shew that
(6.) S.q^=S^q-3SqT^rq,
(e.) a^p^y^+S^al3y = r\afiy,
(d.) S{r.a^yF.Pyar.yal3) = 4: Sa^S^ySyaS.a^y,
(e.) r.q^= (3 S^q-T'' Vq) Yq,
(/.) qVYq-^ = -Sq.Urq + TFq;
and interpret each as a formula in plane or spherical trigonometry.
11. If g- be an undetermined quaternion, what loci are repre-
sented by
(a.) {qa-^r = -a^
{b.) {qa-^Y=a\
{e.) S.{q-aY=a\
where a is any given scalar and a any given vector ?
12. If ^ be any quaternion, shew that the equation
is satisfied, not alone by Q,= ±q but also, by
Q = + ^/~:^{Sq.JJVq-TYq).
(Hamilton, Lectures, p. 673.)
13. Wherein consists the difference between the two equations
T^^=l, and (^^=-1?
a ^a'
What is the full interpretation of each, a being a given, and p an
undetermined, vector ?
14. Find the full consequences of each of the following groups of
equations, both as regards the unknown vector p and the given
vectors a, /3,y:— „ „ on
Sap = 0, Sap = 0,
(«•) of'' " I' (*•) ^•''^P = '^' ^'-^ ^•"^'' = ^'
S.pyp = 0; g^^ ^Q. S.a^yp = 0.
15. From §§ 74, 109, shew that, if e be any unit-vector, and m
any scalar, c" = cos — + e sm — •
70 QUATERNIONS.
Hence shew that if a, j3, y be radii drawn to the corners of a tri-
angle on the unit-sphere, whose spherical excess is m right angles,
/3 + y'a+/3'y + a
Also that, if A, B, C be the angles of the triangle, we have
i£ iB iA
y" ^"a" = — 1.
16. Shew that for any three vectors o, j3, y we have
{Ua^)^ + {UpY)'^+{Uayy + {U.a^yy + iUay.SUa^SUpy = -2.
(Hamilton, Elements, p. 388.)
17. If «i, Og, ag, OS be any four scalars, and p-^, p^, pg any three
vectors, shew that
{8.p^^P^f+{^.a^rp^P^y+ic^{-^rp,p,f-x\^.a^{p,-p,)y
+ 2n(aj2 + Spyp^ + ttjO,^ = 2n(a!2 -f p^) + 2n«''
+ 2{(a!2 +%'' +Pi^) ((^p^pg)'' + 2 «A(aj2 + Sp^pg) -a!^(p2-ps)^)} ;
where Yla^ = a^a^a^.
Verify this formula by a simple process in the particular case
«j = 02 = 03 = a; = 0.
{Ibid)
CHAPTER IV.
DIFFERENTIATION OF QUATERNIONS.
128.] In Chapter I we have already considered as a special case
the differentiation of a vector function of a scalar independent
variable: and it is easy to see at once that a similar process is
applicable to a quaternion function of a scalar independent variable.
The differential, or differential coefficient, thus found, is in general
another function of the same scalar variable ; and can therefore be
differentiated anew by a second, third, &c. application of the same
process. And precisely similar remarks apply to partial differentia-
tion of a quaternion function of any number of scalar independent
variables. In fact, this process is identical with ordinary differ-
entiation.
129.] But when we come to differentiate a function of a vector,
or of a quaternion, some caution is requisite ; there is, in general,
nothing which can be called a differential coefficient ; and in fact
we require (as already hinted in § 33) to employ a definition of a
differential, somewhat different from the ordinary one but, coinciding
with it when applied to functions of mere scalar variables.
130.] If r=F{q) be a function of a quaternion q,
d^ = dFq = ^^n {F{q + '^±)-F{q)},
where » is a scalar which is ultimately to be made infinite, is defined
to be the differential of r or Fq.
Here dq may be any quaternion whatever, and the right-hand
member may be written /., g s
where / is a new function, depending on the form of F; homo-
geneous and of the fi,rst degree in dq ; but not, in general, capable
of being put in the form f ^^) j^_
7 2 QUATERNIONS. [ 1 3 1 .
131.] To make more clear these last remarks, we may observe
that the function y/„ g^
thus derived as the differential of V{q), is distributive with respect
to dq. That is y (^^ ^ + ,) = y (^, ^) + y (^, ,)^
r and « being any quaternions.
For /(?, r + *) = ^^ « (i? (^ + ^) - i^-C?))
And, as a particular case, it is obvious that if a; be any scalar
/fe <»r) = isfiq, r).
132.] And if we define in the same way
dF{q,r,s )
as being the value of
■C.«|'(s+*' '+*••+*• )-^(^.'.'. )}■
where q,r,Sy... dq, dr, ds, are any quaternions whatever ; we
shall obviously arrive at a result which may be written
f{q, r, s, ...dq, dr, ds, ),
where ./ is homogeneous and linear in the system of quaternions
dq, dr,ds, and distributive vrith respect to each of them. Thus,
in differentiating any power, product, &c. of one or more quater-
nions, each factor is to be differentiated as if it alone were variable ;
and the terms corresponding to these are to be added for the com-
plete differential. This differs from the ordinary process of scalar
differentiation solely in the fact that, on account of the non-com-
mutative property of quaternion multiplication, each factor must in
general be differentiated in situ. Thus
d{gr) = dq.r + qdr, but not generally = rdq + qdr.
133.] As Examples we take chiefly those which lead to results
which will be of constant use to us in succeeding Chapters. Some
of the work will be given at full length as an exercise in quaternion
transformations.
(1) {Tpf=-p^.
The differential of the left-hand side is simply, since Tp is a scalar,
2TpdTp.
1 3 3- J DltFEEENTIATION. 73
That Of p^ is ^^n((^p + ±f -p^)
= 2Spdp.
Hence Tp dTp = -Spdp,
or dTp=-S.Updp = sf''
dTp ^dp
or -=i- = ;iS —
Up'
(2) Again, p = TpUp
dp = dTp.Up + TpdUp,
, dp dTp dUp
whence JL-iLj^i^
p Tp Up
= .i + f by(.).
Hence dUp _ -p-dp
W~ J'
This may be transformed into F-^ or -^-^ » &e.
p2 Tp^
(3) iTqy = qKq
2TqdTq = i(^X^) = ^^n^(q + ^J)K(q + ^) -qKq],
= l.-(&±Mi^^,^Kdq),
= qKdq + dqKq,
= qKdq + K{qKdq) (§55),
= iS.qKdq = iS.Kqdq.
Hence dTq = S.UKqdq = S.Uq-'^dq
since :Z^ = :?'% and 27X^ = ?7^-i.
If 3' = p, a vector, Kq = Kp = —p, and the formula becomes
dTp = —S. Updp, as in (1).
Again,
dTq dq
Tq-q
But
dq=TqdUq+UqdTq,
which gives
dq dTq dUq
q- Tq^ Uq'
whence, as
dq_dTq
q Tq
we have
dq _ dUq
f —^
i Uq
74 QUATERNIONS. [134.
2
(4) aif)=<..^(ii+^y-f)
= qdq + dq.q
= 2S.qdq + 2Sq.Fdq + 2Sdq.Vq.
If g' be a vector, as p, Sq and Sdq vanisli, and we have
d{p^) = 28pdp, as in (1).
(5) Let q = r*.
This gives dr^ = dq. But
^ = d{q^) = qdq + dq.q.
This, multiplied iy ^ and m^o Kq, gives
and drKq = dq.Tq^+qdq.Kq.
Adding, we have
qdr + dr.Kq = {q^ + Tq^ + 2<%.j) <«j ;
whence dq, i. e. <^^, is at once found in terms of dr. This process
is given by Hamilton, Lectures, p. 628.
(6) qq-^ = 1,
qdq~^ + dq.q~^ = ;
. • . dq-"^ = — q-^ dq.q-^.
If gf is a vector, = p suppose,
dq~^ = —p~^dp.p~^
p^ p p
(7) q = Sq+Fq,
dq= dSq + dFq.
But dq = ^^j- + Fi:?^.
Comparing, we have
dSq = Sdq, dVq = Vdq.
Since Xq = Sq— Vq, we find by a similar process
<?X2 = Kdq.
134.] Successive diflFerentiation of course presents no new dif-
ficulty.
Thus, we have seen that
d{q^) = dq.q + qdq.
1 35-] DIFFEEENTIATION. 75
DiflFerentiating again, we have
and so on for higher orders.
If §' be a vector, as p, we have, §133(1),
d{p^) = 2Spdp.
Hence d^(p^) = 2{dpf + 2Spd^p, and so on.
Similarly d^Up= -dA-Fpdp) •
But d
<Tp^
1 _ 2dTp 2Spdp
Tp"^ ~ Tp^ ~ Tp*^
and d. Vpdp = V. pd^p.
Hence -^^J^p =- ^(rpi,)^+ Wp ^ 2J^^^^
= - ^ ((^P^P)' +P' Fp^V- 2 Fp^p^p^p) *
135.] If the first differential of q be considered as a constant
quaternion, we have, of course,
d^q = 0, d^q = 0, &e.,
and the preceding formulae become considerably simplified.
Hamilton has shewn that in this case Taylor's Theorem admits of
an easy extension to quaternions. That is, we may write
f{q + xdq) =/{q) + xd/{q) + ~ d^iq) +
if d'^q = ; subject, of course, to particular exceptions and limita-
tions as in the ordinary applications to functions of scalar variables.
Thus, let y($') = q^) and we have
4f(q) = q^dq + qdq.q + dq.q^,
d^/iq) = 2dq.qdq + 2q{dq)^ + 2idq)''q,
d^f{q) = G{dq)\
and it is easy to verify by multiplication that we have rigorously
(g- + xdqf= f + x{q^dq + qdq.q + dq.q^) + x" {dq.qdq 4 q {dqf + {dqfq) + a;^(dqf ;
which is the value given by the application of the above form of
Taylor's Theorem.
As we shall not have occasion to employ this theorem, and as the
demonstrations which have been found are all too laborious for an
elementary treatise, we refer the reader to Hamilton's works, where
he will find several of them.
* This may be farther simplified ; but it may be well to caution the student that
we cannot, for such a purpose, write the above expression as
-^J.pidpYpdp + d'p.p-'- 2dpSpdp}.
76 QUATERNIONS. [136.
1 36.] To differentiate a function of a function of a quaternion
we proceed as with scalar variables, attending to the peculiarities
already pointed out.
137.] A case of considerable importance in geometrical appli-
cations of quaternions is the differentiation of a scalar function of p,
the vector of any point in space.
Let F{p) = C,
where i^ is a scalar function and C an arbitrary constant, be the
equation of a series of surfaces. Its differential,
f{p, dp) = 0,
is, of eourscj a scalar function : and, being homogeneous and linear
in dp, § 130, may be thus written,
Svdp = 0,
where i; is a vector, in general a function of p.
This vector, v, is easily seen to have the direction of the normal
to the given surface at the extremity of p ; being, in fact, per-
pendicular to every tangent line dp, §§ 36, 98. Its length, when F is
a surface of the second degree, is as the reciprocal of the distance of
the tangent-plane from the origin. And we will shew, later, that if
p = ix+jy+&z,
/ . d . d , d \ „
EXAMPLES TO CHAPTER IV.
1 . Shew that
(a.) d.SUq = s.Usr^=-s^Truq,
(b.) d.rUq=r.Uq-^F^dq.q-^),
(c.) d.TrUq = S^=:S^^SUq,
{d.) d.a" = ^ a^+'^dm,
(e.) d\Tq={^.dqq-^-S.{dqq-^f}Tq = -~Tqr^^'
2. If Fp='2.Sap8l3p+iffp^
give dFp t= Svdp,
shew that v = S T. ap^ + (^ + 2 Sa^) p.
CHAPTER V.
THE SOLUTION OP EQUATIONS OF THE PIEST DEGEEE.
138.] We have seen that the differentiation of any function
whatever of a quaternion^ q, leads to an equation of the form
where/" is linear and homogeneous in dq^. To complete the process
of differentiation, we must have the means of solving this equation
so as to be able to exhibit directly the value of dq.
This general equation is not of so much practical importance as
the particular case in which dq is a . vector ; and, besides, as we
proceed to shew, the solution of the general question may easily be
made to depend upon that of the particular case j so that we shall
commence with the latter.
The most general expression for the function _/ is easily seen to be
dr =/(§■, dq) = 2 V.adqh + S.cdq,
where a, I, and c may be any quaternion functions of q whatever.
Every possible term of a linear and homogeneous function is re-
ducible to this form, as the reader may easily see by writing down
all the forms he can devise.
Taking the scalars of both sides, we have
Sd^- = S.cdq = SdqSa + S.rdqFc.
But we have also, by taking the vector parts,
Fd?- = 2 r. adqb = Sdq.^ rab + -2,r.a{ Vdq) b.
Eliminating Sdq between the equations for Sdr and Vdr it *is
obvious that a linear and vector expression in Vdq will remain.
Such an expression, so far as it contains Vdq, may always be reduced
to the form of a sum of teims of the type aS.^Vdq, by the help of
formula like those in §§ 90, 91. Solving this, we have Tdq, and
Sdq is then found from the preceding equation.
78 QUATERNIONS. [139.
139.] The problem may now be stated thus.
Find the value of p from the equation
o5/3p+ai-S)3ip+ ... = 2.aSfip = y,
where a, 13, a^, ^i, ...y are given vectors. [It will be shewn later
that the most general form requires but three terms, i. e. six vector
constants a, y3, a^, ^j, Og, /Sg in all.]
If we write, with Hamilton,
(j>p = 2.a<S)3p,
the given equation may be written
<pp = y.
or p = (j>-^y,
and the object of our investigation is to find the value of the in-
verse function (jr'^,
140.] We have seen that any vector whatever may be expressed
in terms of any three non-coplanar vectors. Hence, we should ex-
pect a priori that a vector such as <p(p4>p, or <j)^p, for instance, should
be capable of expression in terms of p, <j)p, and (p^p. [This is, of
course, on the supposition that p, (j)p, and (fi^p are not generally co-
planar. But it may easily be seen to extend to this case also. For
if these vectors be generally coplanar, so are <j)p, (p^p, and <j)^p, since
they may be written <r, ifxr, and (/)V. And thus, of course, ^^p can
be expressed as above. If in a particular case, we should have, for
some definite vector p, <pp=gp where ^ is a scalar, we shall obviously
have <^^p =g^p and ^^p =g^p, so that the equation will still subsist.
And a similar explanation holds for the particular case when, for
some definite value of p, the three vectors p, Kpp, <^^p are coplanar.
For then we have an equation of the form
^^p = Ap-i- Bijip,
which gives (l>^p = A(l>p + £(l)^p
= ABp-\-{A + B^)<i>p.
So that (p^p is in the same plane.]
If, then, we write
-(t,^p = xp+y4>p + e(l)^p, (1)
it is evident that x, y, z are quantities independent of the vector p,
and we can determine them at once by processes such as those iu
§§91,92.
If any three vectors, as «', /, h, be substituted for p, they will in
general enable us to assign the values of the three coeflScients on
142.] SOLUTION OF EQUATIONS, 79
the right side of the equation, andme solution is complete. For
by putting (t>~^p for p and transposing, the equation becomes
that is, the unknown inverse function is expressed in terms of direct
operations. If x vanish, while y remains finite, we substitute ^~V
for p, and have -y (^-^ = «p + cj,p,
and if x and _y both -vanish
— Z(j>~^p = p.
141.] To illustrate this process by a simple example we shall
take the very important case in which <f) belongs to a central surface
of the second order ; suppose an ellipsoid ; in which case it will be
shewn (in Chap. VIII.) that we may write
^p = —a^iSip — h^jSjp—c^JcSkp,
Here we have
ipi = cp'i, <^H = aH, <f)H = a^i,
4,j = by, <t>y = b*j, <t>y = by,
(pk = cH, ^H = c*/i, ^^k = o^h.
Hence, putting separately i,j, Tc for p in the equation (1) of last
section, we have —a^ = x^ya^-\-m^,
—b^= le+yb^+zb*,
— C® = X-\-i/C^ +ZC*.
Hence a^, b^, c^ are the roots of the cubic
^* + «P +.?'£+«= 0,
which involves the conditions
z=-{a^ + l^ + c^),
y = cfib"^ + b'^c^ + c^a^,
x = — a^b^c^.
Thus, with the above value of ^, we have
(/>3p = aWc^p - {aW + h^c^ + c V) # + {a^ + b^-\- c^) <p^p.
142.] Putting ^"^(T in place of p (which is any vector whatever)
and changing the order of the terms, we have the desired inversion
of the function ^ in the form
aWc^-'^a- = {aW + bH^ + (^a^) a—{a'^ + b^ + c^) (fxr + ^V,
where the inverse function is expressed in terms of the direct func-
tion. For this particular case the solution we have given is com-
plete, and satisfactory; and it has the advantage of preparing the
reader to expect a similar form of solution in more complex cases.
80 QUATERNIONS. [143.
143.] It may also be useful as a preparation for what follows, if
we put the equation of § 141 in the form
= *(p') = 4,^p-{a^ + 6^ + c^)(l>''p + {aH'^ +¥c^ +c^a^)^p-a%^c^ p
= {(«^-«') (</>-*') (<^-«')}p (2)
This last transformation is permitted because </> is commutative with
scalars like a*, i. e. <p{a^p) = a^^p.
Here we remark that (by § 140) the equation
r.p0p = 0, or ^p = gp,
where g is some undetermined scalar, is satisfied, not merely by
every vector of null-length, but by the definite system of three rect-
angular vectors Ai, Bj, Ck whatever be their tensors, the corre-
sponding particular values of g being a^, h^, c^.
144.J We now give Hamilton's admirable investigation.
The most general form of a linear and vector function of a vector
may of course be written as
</)p = 'S.V.qpr,
where q and r are any constant quaternions, either or both of which
may degrade to a scalar or a vector.
Hence, operating by S.a- where o- is any vector whatever,
S(r(l>p = 2ScTF.qpr = '28pF.raq = 8p4)'(T, (3)
if we agree to write ^'o- = IiF.raq,
and remember the proposition of § 88. The functions <^ and <j/ are
thus conjugate to one another, and on this property the whole in-
vestigation depends.
145.] Let A, p. be any two vectors, such that
^p ^ Vkp,.
Operating by SX and S.p. we have
8k<^p = 0, Sp.(t>p = 0.
But, introducing the conjugate function <^', these become
Sp(f>'K = 0, Sp^'p. = 0,
and give p in the form mp = Fcjt'kcli'p,,
where mis a scalar which, as we shall presently see, is independent
of A, jM, and p.
But our original assumption gives
p = <(>-W\ix;
hence we have m^~Wkp. = F^'k(p' p., (4)
and the problem of inverting <^ is solved.
147.]
SOLUTION OF EQUATIONS.
81
146,] It remains to find the value of the constant m, and to
express the vector Vd/kcb'u
as a function of FX/n.
Operate on (4) by /S.^'r, where v is any vector not coplanar with
X and /n, and we get
mS.(j/v(l>-^F\n = mS.v<i><irWKix (by (3) of § 144)
= mS.Kixv = S.^'X^'ji^'v, or
m =
S.Xfxv
(5)
J3 q
r
S.4>'K4>'iJ.(^'v,
Pi ix
^1
H 2'2
^i
p q
r
S.KjxVf
Pi Si
'•i
[That this quantity is independent of the particular vectors \, ju, v
is evident froija the fact that if
k'=p)\. + qiJL + ri>, i/ = pjk + q-i^ix + r.^v, and d'= j?2^+S'2M+»"2»'
be any other three vectors (which is possible since X, [x, v are not
coplanar), we have
<i)'k'= p<i)KJrq^' !!.-{- r<l>'v, &C., &C.-,
from which we deduce
and
so that the numerator and denominator of the fraction which ex-
presses m are altered in the same ratio. Each of these quantities
is in fact an Invariant, and the numerical multiplier is the same for
both when we pass from any one set of three vectors to another.
A still simpler proof is obtained at once by writing A +j3/x for \
in (5), and noticing that neither numerator nor denominator is
altered.]
147.] Let us now change ^ to <i>-\-g, where g is any scalar. It
is evident that ^' becomes <i>'+g, and our equation (4) becomes
mg{4>^-g)-WkiJ,= r{4,''+g)k{<t>'+g)ixi
= r<t>'k^'ix+gF((l/k,j. + k<t>',x)+g'rk^,
= {'m(t)~^ +gx+g^)V^kix suppose.
In the above equation
_ S.{cl>',+g)k{ct/+g)t,{^'+g)v
'^'- sJili,
= m+m^g+m^g^+g^
82 QUATERNIONS. [148.
is what m becomes when ^ is changed into ^-Vg; % and m^ being
two new scalar constants whose values are
"^ Sl^v '
_ S. {kij.(f>'v + 4>' kfiv + X.(l>'iJLv)
If, in these expressions, we put k+pjx for \, we find that the terms
in jp vanish identically ; so that they also are invariants. Substi-
tuting for Mg, and equating the coefficients of the various powers
of ^ after operating on both sides by ^-f-^, we have two identities
and the following two equations,
% = '^ + X.
[The first determines x, and shews that we were justified in treat-
ing F{((/\ij,-\-\<f>'^i) as a linear and vector function of F.Xi/,. The
result might have been also obtained thus,
SAx^Xfi. = S.\<f/\ix=—S.\ix(t/\=-8.\(l)rhiJ.,
S.fjLx^^fJ^ = S.jjlKcj/ij, = —S.iiipVKjj,,
S.vxVXix. = S.{v^'Xii. + vk4>'i).)
= m2SKij,v—S.\iJi.^'v
= S.v {m^Vkfi—^fKii) ;
and all three (the utmost generality) are satisfied by
X = %- *-J
148.] Eliminating ^ from these equations we find
or m<l)~^ = OTj — ^j (^ 4- (/)^,
which contains the complete solution of linear and vector equations.
149.] More to satisfy the student of the validity of the above
investigation, about whose logic he may at first feel some diffi-
culties, than to obtain easy solutions, we take a few very simple
examples to begin with : we treat them with all desirable prolixity,
and we append for comparison easy solutions obtained by methods
specially adapted to each case.
150.] Example I.
Let <l>p = V.apfi = y.
Then <^'p = V.^pa = <^p.
Hence m = -=r^ — S ( V. aX^ V. au/3 V. av^).
8.\iJ,v ^ '
1 5 3-] SOLUTION OP EQUATIONS. 83
Now X, n, V are any three non-eoplanar vectors; and we liiay
therefore put for them a, ^,y if the latter be non-coplanar.
With this proviso
% = -s-^'Sf(a2/3.a/3'2.y + a.a/32.r.oy^ + a2/3.j3.r,ay/3)
,2o2
= — O'
1
S.a^y
= —Sap.
S (Ti^^.yS.y + a.a/32.y + o;8 V.ayfi)
Hence
which is one form of solution.
By expanding the vectors of products we may easily reduce it to
the form a^^^Safi.p = - a^/S^ y + a^^Say + Ba^Sfiy,
a-^Say + B-^S3y—y
or p = — ' ■ — -^ — - ■
151.] To verify this solution, we have
^•"''^ "= ^O-^ay + a-^/Sy-r.ay/S) = y,
which is the given equation.
153.] An easier mode of arriving at the same solution, in this
simple case, is as follows : —
Operating by S.a and S.p on the given equation
r.opjS = y,
we obtain a^SjSp = Say,
P^Sap = S^y ;
and therefore aSfip = a~'^Say,
pSap = /8-i/S'/3y.
But the given equation may be written
aS^p—pSa^ + ISSap = y.
Substituting and transposing we get
pSafi = a'^Say + p-^S^y—y,
which agrees with the result of § 150.
153.] If a, fi, y be coplanar, the above mode of solution is appli-
cable, but the result may be deduced much more simply.
For (§101) S.aPy = 0, and the equation then gives S.a^p = 0, so
that p is also coplanar with a, /3, y.
6 3,
84 QUATEENIONS. [154.
and at once „ _ „-i,,o-i
Hence the equation may be written
app = y,
P = a"V^'
and this, being a vector, may be written
This formula is equivalent to that just given, but not equal to it
term by term. [The student will find it a good exercise to prove
directly that, if o, /3, y are coplanar, we have
^(a-i/Sfay + ^-i*S/3y-y) = a-'^S^'^y^^-^Sar'^y-ySar'^^-'^r^
The conclusion that o a n
b.app = 0,
in this case, is not necessarily true if
5a/3 = 0.
But then the original equation becomes
aS^p + pSap =: y,
which is consistent with
S.aPy = 0.
This equation gives
^("'^-^«^) = «U/y ^A + ^ Say S '
by comparison of which with the given equation we find
Sap and S^p.
The value of p remains therefore with one indeterminate scalar.
154. J Example II.
Let <^p =: V.a^p = y.
Suppose a, ;8, y not to be coplanar, and employ them as A, ft, v to
calculate the coefficients in the equation for (j)"^. We have
S.(T(j)p = S.cra^p = S.pKcra^ = S-pcj/a:
Hence <^'p = ^-palS = V.I3ap.
We have now
= a^fi^Safi,
m, = -=-— -(5(0.^0/3. r, /Say + ;3a2./3.r.;3ay + ;3a2./3a/3.y)
o.apy
= 2{Safif + a^^^,
«*2 = "cV S(a.^.r.fiay+a.^a^.y + ^aK^.y)
o.apy
= 38al3.
156.] SOLUTION OF EQUATIONS. 85
Hence
= (2 (<So^)2 + a^j3^) y- 3/Sa/3 V.a^y + V.a^ V.a^y,
which, by expanding the vectors of products, takes easily the simpler
form a^p2Sa^p ^ a^^2y_^^2s^^ ^ 2^Sa^Say-^a?S^y.
155.] To verify this, operate by F.a/3 on both sides, and we have
a^^^Sapr.aPp = a'^^W.afiy- r.a^afi'^Say+2ap^Safi8ay-ao?^'^S^y
= a?^^ {a8^y-pSay + ySaP)-{2aSap-^a^)P^Say
+ 2 afi^Sa^Say—aa^^^S^y
= a^p^Sa^.y,
or V. afip = y.
156.] To solve the same equation without employing the general
method, we may proceed as follows : —
y = r. a^p = pSa^ + V. r{a^) p,
Operating by S. Fa^ we have
S.a^y — S.a^pSa^.
Divide this by Sa^, and add it to the given equation. We thus
obtain o o
y + ^^ =pSal3+ r. Viafi) p + S. r{afi) p,
= {Sal3+ral3)p,
= a/3p.
Hence p = /3-1 a-i (y + -^) ,
a form of solution somewhat simpler than that before obtained.
To shew that they agree, however, let us multiply by a^^^Safi,
and we get a^/i^Sa^.p = ^aySa^ + fiaS.a^y.
In this form we see at once that the right-hand side is a vector,
since its scalar is evidently zero (§ 89). Hence we may write
a?^^Sa^.p = r.^aySa^-Va^S.a^y.
But by (3) of §91,
—yS.ap ra/3 + a/S./S ( Fa^) y + /3/S. F{aP) ay + Fa^S.a^y = 0.
Add this to the right-hand side, and we have
a^P^Sa^.p = y {{Sa^)^-S.al3Fap)-a {Sa0S^y-^S.^ (Fafi) y)
+ ;8 {Sal3Say + S.F (afi) ay) .
But {Safif-S-a^Fa^ = {Sa^f-{Fa^f = a^^\
Sa^8^y-S.fi{Fa^) y = Sa^Sfiy-SftaS^y + ^^Say = ^^Say
SapSay + S.F{aP)ay = SafiSay + SafiSay-a^S^y
= 2Sa^Say-a'^S^y;
and the substitution of these values renders our equation identical
with that of § 154.
86 QUATERNIONS. [157.
[If n, /3, y be coplanar, the simplified forms of the expression for p
lead to the equation
Sap.p-^a-^y = y-a-^Say + 2pSa-^fi-^Say-p-'8l3Y,
which, as before, we leave as an exercise to the student.J
157.] Example III. The solution of the equation
Tep = y
leads to the vanishing of some of the quantities m. Before, how-
ever, treating it by the general method, we shall deduce its solution
from that of V.a^p = y
already given. Our reason for so doing is that we thus have an
opportunity of shewing the nature of some of the cases in which one
or more of m, m^, m^ vanish; and also of introducing an example
of the use of vanishing fractions in quaternions. Far simpler solu-
tions will be given in the following sections.
The solution of the last- written equation is, § 154,
a^^^Sa^.p = a^^^y-a^^Say—^a.'^S^y+2^Sa^8o.y.
If we now put o^ = e + e
where e is a scalar, the solution of the first-written equation will
evidently be derived from that of the second by making e gradually
tend to zero.
We have, for this purpose, the following necessary transforma-
tions : - a2^2 _ „^ x.a^ = (e + e) (e - e) = e^ - e^,
a^^Say + ^a?8^y = a^.^Say + jSa.aS^y,
= {e + e)fiSay + {e—e)aSPy,
= e {^Say + aS^y) + eV.yVa^,
= e l^Say + aS^y) + e Tye.
Hence the solution becomes
(e2_e2)ep = {e^-e.^)y-e{^Say + aS^y)-iryi + 2e^Say,
- le^—(^)y + eF.yra^—eryf,
= ^e^^i'')y + ery€ + yf'-fSyf,
= e^y + eVye — tSye,
Dividing by e, and then putting e = 0, we have
-eV = rye-<„(^).
Now, by the form of the given equation, we see that
Sye = 0.
Hence the limit is indeterminate, and we may put for it cc, where as
is any scalar. Our solution is, therefore,
or, as it may be written, since Sye = 0,
p = e-i(y + a;).
l6o.J SOLUTION OF EQUATIONS. 87
The verification is obvious — for we have
ep = y + x.
158.] This suggests a very simple mode of solution. For we
see that the given equation leaves Sip indeterminate. Assume,
therefore, Sep = x
and add to the given equation. We obtain
ep =x + y,
or p=e-i(y + a,),
if, and only if, p satisfies the equation
Vep = y.
159.] To apply the general method, we may take e, y and ey
(which is a vector) for A, p,, v.
We find <l)'p = Vpe.
Hence
Hence
m = 0,
1-
^l=-^,S.{..,y.,^y)=-,^
m„ = 0.
or
That is.
P
= e~^y + xe, as before.
Our warrant for putting xe, as the equivalent of 0"^ is this : —
The equation ^2^ _ q
may be written r.eFfcr = = <re^ - tSta.
Hence, unless o- = 0, we have o- 1| e = xe.
160.] Example IV. As a final example let us take the most
general form of t^, which, as will be soon proved, may be expressed
as follows : —
<^p = ajS/3p + a-^S^-^p + a^S^^p = y.
Here <l>p = ^Sap + ^-^80^ p + ^.^Sa^ p,
and, consequently, taking a, Oj , 02, which are in this case non-
coplanar vectors, for A, p., v, we have
-S.(^Saa + ^^Sa^a + P^Sa^oi) {l3Saa-^ + P-^Saj^aj^+ ...) {pSaa2+ )
'S.
aojOg
S.1313,^2
Saa Sa-jO, iSogO
Saor^ /SojOj (SogOj
Saag /iSa^ag Sa^a^
S.aa^a^
= ^f^{ASaa + A^Sa^a + A,Sa^a),
'S.o
88 QUATERNIONS. [l6o.
where A = Sa^aiSa^ai— Sai.aiSaia2
= —S. VojO^ VojO^
A^ = Sa^aiSaa2—Saa,ySa^a3
= —S. Fctg a FioiOg
A2 = SacijSa^a2 — Saia^Saa^
r= — S. Vaa^ FajOg.
Hence the value of the determinant is
— {SaaS. FojO^ Va^a^ + SoyoS. Fa^a Va^a^ + Sa^aS. Faa^ FoiOg)
= -SMiFaia^S.aaja^) {by § 92 (3)} = -{S.aa^a^f.
The interpretation of this result in spherical trigonometry is very
interesting, (^ee Ex. (6) p. 68.)
By it we see that
Similarly,
m, = g 8.[a (0Saai + ^,80.0, + /SgSoaai) {^Saa^ + ^i^ojOa + /SaiSaga,) + &c.]
a.aOiO^
= 5 {S.a^Pi (800180102— 8010^8002) + )
= o (5.a/3i3i5.a r.Oi ragOi + )
= - ^ IS.a ( FpPy8. Faoi Foy02 + F^^8. F02O rajOg + Ffi^^2^. Fo^a^ Fa^a^
tf.aoia^ +S.ai{Fl3l3^8.FaoyFo20+ )
+ S.a2{Fl3Pi8.FaoyFooi+ )] ;
or, taking the terms by columns instead of by rows,
= — p 18. F^Pi {a8. Faoy Fa^a^ + 0^8. Faa^ Fa^ + 028. Faa-^ Faa-^
8.00^02 1
= --^^^lS.Ffi^y{FoayS.oay02) + ],
_ = -S{FaoiFpfii+ Fai02F^,p2+ T^V^^^i^)-
Again,
«2„ = -r S\oaA^Saa2 + ^iSoia2+ ...) + a2o{^8oai-\- ...) + a-^02{pSaa+...)'\,
or, grouping as before,
= — 8\^{ Foai8ao2 + Fa.^o8aay + Foy028aa) 4- • ■ ■] ,
0.00^02
= -^^^S[fi {08.00^02)+ j (§92(4)),
= 8(0^+0^^1+02^2)-
And the solution is, therefore,
(f>-^y8.aOya28.p^l^2 = pi^.aaiaa&^^i^a
= y25. Fooy T/S/Sj + ^ySSo^S - <J) V-
163-] SOLUTION OF EQUATIONS. 89
\ [It will be excellent practice for the student to work out in detail
the blank portions of the above investigation, and also to prove
directly that the value of p we have just found satisfies the given
equation.]
161. J But it is not necessary to go through such a long process
to get the solution — though it wUl be advantageous to the student
to read it carefully — ^for if we operate on the equation by S-OjCt^,
S.a^a, and Smo^ we get
S.aiOf^aSlSp = S.aiO^y,
S.a^aoiSfiip =: S.a^ay,
S.aayO^S^^p ^ S.aajy.
From these, by § 92 (4), we have at once
pS.aojO^S.pPiP^ = Fpi3^8.aajy+ r^^^S.a^a^y + V^^S.a^ay.
The student will find it a useful exercise to prove that this is equi-
valent to the solution in § 160.
To verify the present solution we have
= a'S'.;8/3ij32iS'.aj^a2y-|-aj^&;8j^^2/3'^-"2'*y + ''2'^-/32/3/5i'^-°"iy
= S.^l3,fi^ {yS.aaia,), by § 91 (3).
163.J It is evident, from these examples, that for special cases
we can usually find modes of solution of the linear and vector equa-
tion which are simpler in application than the general process of
§ 148. The real value of that process however consists partly in
its enabling us to express inverse functions of 0, such as ((^+^)~^
for instance, in terms of direct operations, a property which will be
of great use to us later ; partly in its leading us to the fundamental
cubic ^^—m^^^ +mj(f>—m = 0,
which is an immediate deduction from the equation of § 148, and
whose interpretation is of the utmost importance with reference to
the axes of surfaces of the second order, principal axes of inertia,
the analysis of strains in a distorted solid, and various similar
enquiries.
163.] When the fiinction <^ is its own conjugate, that is, when
Spcpo' = Scrcfip
for all values of p and o-, the vectors for which
{<t>-ff)p =
form in general a real and definite rectangular system. This, of
course, may in particular cases degrade into one definite vector, and
any pair of others perpendicular to it ; and cases may occur in
which the equation is satisfied for every vector.
90 QUATERNIONS. [164,
Suppose the roots oi mg= (§ 147) to be real and different, then
VPi — ffiPi 1 ^j^ere p^, p^, P3 are three definite vectors determined
Wi — y2P2 f jjy. ^jijg constants involved in <ji.
<t>Pa =ffsP3'
Hence ^i^2%P2 = '5-M#2
= S.pT,4>^P2, or = S.p^ip^pi,
because ^ is its own conjugate.
But (^^2 = fflPz)
<l>^Pi=ffiPi>
and therefore 9x9i^P-iPi = 9l^PiP2 = ^fi^f 1P2 >
which, as g^ and g^ are by hypothesis different, requires
SP\P2 = 0-
Similarly 'S/'2P3 = 0, 'S'pgpj^ = 0.
If two roots be equal, as g^, g^, we still have, by the above proof,
iSpiPg = and Sp^p^ = 0. But there is nothing farther to determine
/>2 and P3, which are therefore any vectors perpendicular to py
If all three roots be equal, every real vector satisfies the equation
(<^_(/)p=0.
164.] Next, as to the reality of the three directions in this case.
Suppose g^-'r^N — 1 to be a root, and let pg + tr^'v— 1 be the
corresponding value of p, where g,^ and ^2 ai'c real numbers, pg and a^
real vectors, and v — 1 the old imaginary of algebra.
Then ^{p^ + cTg a/— 1 ) = (^2 + >^2 ■v^^^) (P2 + "^a v'— ^).
and this divides itself, as in algebra, into the two equations
#2 = ^2^2 — '^2'^2)
(/mJ-2= /?2P2+^2°'2-
Operating on these by /S.o-g, /S.pg respectively, and subtracting the
results, remembering our condition as to the nature of <^
<S'a-20P2 = Sp^^lT^,
we have ^gC"'! +Pi) = 0.
But, as o-g and p^ are both real vectors, the sum of their squares
cannot vanish. Hence h^ vanishes, and with it the impossible part
of the root.
165. J When ^ is self-conjugate, we have shewn that the equa-
^^^"^ g^— m^g^ + m-^g —m — Q
has three real roots, in general different from one another.
Hence the cubic in ^ may be written
{.<i>-9i)i.^-9^{.4>-9s) = 0>
167.] SOLUTION OF EQUATIONS. 91
and in this form we can easily see the meaning of the cubic. For,
let pi, p2, pg be three vectorg such that
{^-ffi)pi = 0. {'t>—ff2)P2 = 0, {<t>—9^Ps = 0.
Then any vector p may be expressed by the equation
pS-PiP2Pa = pAP2P3P + P2.^-P3PiP + Pa^-PiP2P (§91).
and we see that when the complex operation, denoted by the left-
hand member of the above symbolic equation, is performed on p, the
first of the three factors makes the term in pj vanish, the second
and third those in p^ and pg respectively. In other words, by the
successive performance upon a vector of the operations <f> — ^j, (p — ff^'
^—g^, it is deprived successively of its resolved parts in the direc-
tions of Pi, p^, Pg respectively j and is thus necessarily reduced to
zero, since pj, pg, pg are (because we have supposed g-^^, g^, g^ to be
distinct) distinct and non-eoplanar vectors.
166.] If we take pj, pg, pg as rectangular K^zi^- vectors, we have
— p = p-iSpjp + P2,8p2P + Ps'SpaP,
whence # = —SiPx^pT^P—g^p^Sp^p—g^^Sp^p ;
or, still more simply, putting i, j, h for p^, pg, pg, we find that any
self-conjugate function may be thus expressed
^P = —9ii^P —ad^JP —9i^Skp,
provided, of course, i, j, k be taken as roots of the equation
Vp^p = 0.
167.] A very important transformation of the self-conjugate
linear and vector function is easily derived from this form.
We have seen that it involves three scalar constants only, viz. y^,
g , g^. Let us enquire, then, whether it can be reduced to the fol-
lowing form <j)p =/p + AF.{i + e/i:)p{i— eh),
which also involves but three scalar constants/, h, e. Here, again,
i, y, h are the roots of Vp^p = 0.
Substituting for p the equivalent
p = —iSip—jSjp—kSkp,
expanding, and equating coefficients of i,j, k in the two expressions
for <\>p, we find —g^ = —/+ ^^ (2 — 1 -[- e%
-g,=-.f-k{2e'^ + l-e^).
These give at once
-(^1-^2) = 2-^,
-{9z-9z) = Se^/J.
92 QUATERNIONS. [l68.
Hence, as we suppose the transformation to be real, and therefore e^
to be positive, it is evident that ffi — ff^ and ^2 — ffa have the same
sig^ ; so that we must choose as auxiliary vectors in the last term
of <pp those two of the rectangular directions i, j\ k for which the
coefficients g have the greatest and least values.
We have then ^i^9j-h.,
9i-9z
^=-\ {91-92),
and f=\{gi+gi).
168.] We may, therefore, always determine definitely the vec-
tors \, fi, and the scalar y, in the equation
when <\> is self-conjugate, and the corresponding cubic has not equal
roots, subject to the single restriction that
is known, but not the separate tensors of X and fx. This result is
important in the theory of surfaces of the second order, and will be
considered in Chapter VII.
169.] Another important transformation of <^ when self-conju-
gate is the following, ^p = aaVap + i^Sfip,
where a and b are scalar s, and a and /3 unit-vectors. This, of
course, involves sis scalar constants, and belongs to the most gen-
eral form 4)p = —giPiSpiP—g2P2^P2p—9aP3^PaPy
where pi, pg, p^ are the rectangular unit-vectors for which p and (pp
are parallel. We merely mention this form in passing, as it be-
Ipngs to the Jveal transformation of the equation of surfaces of the
second order, which will not be farther alluded to in this work. It
will be a good exercise for the student to determine a, ;8, a and b,
in terms of i^^, yg. 93, ai"i Pi> P2, Pa-
170.] We cannot afford space for a detailed account of the sin-
gular properties of these vector functions, and will therefore content
ourselves with the enuntiation and proof of one or two of the most
important.
In the equation nKp'^FXpi. = F(\>'\<\)'ii. (§ 145),
substitute \ for ^'K and ji for <^'p., and we have
»«rqb'-i\<^'-V = ^FKix.
Change ^ to <p+g, and therefore ^' to <\> +g, and m to %, we have
a formula which will be found to be of considerable use.
1 72-] SOLUTION OF EQUATIOKS. 93
171.] Again, by § 147,
Similarly -^ S.p (</> + ^)- V = t ■^P^" V + ^P + ^P^-
Hence
'^S.pi^+,)-^p-^S.pi^ + A)-^P = i,-,)[p^-'^].
That is, the functions
are identiealj i. e. when equated to constants represent the same series
of surfaces, not merely when
g = h,
but also, whatever be g and h, if they be scalar functions of p which
satisfy the equation mS.p(j>-^p = gkp'^.
This is a generalization, due to Hamilton, of a singular result ob-
tained by the author *.
173.] The equations
S.p((l>+g)-^p = 0,l
S.p{<p + A)-^p=0,i ^ '
are equivalent to mSp(j)~^p+gSp\p+ff^p^ = 0,
mSp<t>-'^p + hSpxp + h^p^ — 0.
Hence m{\—x) Sp4>-''-p + {g—M) Spxp + {g^ — A^(c)p^ = 0,
whatever scalar be represented by x.
That is, the two equations (1) represent the same surface if this
identity be satisfied. As particular cases let
(1) aj=l, in which case
Sp-\p+g + h = 0.
(2) g—hx=0, in which case
m{l - |)^p-i0-V + (/->^^f) = 0,
or mSp~^4>~^P~9^ — "•
(3) a- = |a. giving
m{\- |-,) -Spr V + (^ - >^ fg) *PXP = 0,
or m {A+g)Sp(l>-^p +g/iSpxp = 0.
* Note on the CarteBian equation of the Wave-Surfaee. Quarterly Math. Jowmal,
Oct. 1859.
94 QUATERNIONS. [l73-
173.] In various investigations we meet with the quaternion
J = a<l>a + I3<j>p + y<l>y,
where a, /3, y are three unit-vectors at right angles to each other.
It admits of being put in a very simple form, which is occasionally
of considerable importance.
We have, obviously, by the properties of a rectangular unit-
system ^ _ ^y^a + yatl>l3 + a^<}>y.
As we have also s.afiy = _ i (§71(13)),
a glance at the formulae of § 147 shews that
at least if ^ be self-conjugate. Even if it be not, still (as will be
shewn in § 1 74) ^p = ^'p + r^p,
and the new term disappears in Sq.
We have also, by § 90 (2),
Vq=a{Sfi(l>y-Sy<pp)-i-P{Sy<j>a-Sa4r/) + y{Sa(l>^-Sl3<f>a)
= a8fi{4)~<l/)y + fiSy{(t>-<tt)a + ySa{(l>—^')^
= aS.fiey + ^S.yea+ yS.aep
= — {aSae + /S/S/Se + ySye) = e.
[We may note in passing that this quaternion admits of being
expressed in the remarkable form
where V = OT-+S-7-+y-5->
ax ay dz
and p=ax-\-^y-\-yz.
We will recur to this towards the end of the work.]
Many similar singular properties of <\> in connection with a rect-
angular system might easily be given ; for instance,
V{a F<l>^(j>y + j3 Ffycl)a + y F^a<p^)
= mF(a<j)-^a + fi^'~^fi + y^'-'^y) = mF.Vi^'-'^p = 4>e ;
which the reader may easily verify by a process similar to that just
given, or (more directly) by the help of § 145 (4). A few others
will be found among the Examples appended to this Chapter.
174.] To conclude, we may remark that as in many of the
immediately preceding investigations we have supposed <f> to be
self-conjugate, a very simple step enables us to pass from this to
the non-conjugate form.
For, if ^' be conjugate to (^, we have
Sp(j>'(r = 8<T<pp,
and also Spt^xr = Sa-^'p.
17 7-] soLUTioi^r OF equations. 95
Adding, we have
SO that the function {<f> + <j)') is self-conjugate.
Again, Sp(f>p = Spcj/p,
which gives Sp{^—<^')p =. 0.
Hence {<f>-~<l)')p = Fep,
where, if ^ be not self-conjugate, e is some real vector, and therefore
<t>P = \{<l> + <l>')p+\{4>-^')p
=U<P + <t>')p+hrip.
Thus every non-conjugate linear and vector function differs from
a conjugate function solely iy a term of the form
Fep.
The geometric signification of this will be found in the Chapter on
Kinematics.
175.] We have shewn, at some length, how a linear and vector
equation containing an unknown vector is to be solved in the most
general case ; and this, by § 138, shews how to find an unknown
quaternion from any sufficiently general linear equation containing
it. That such an equation may be sufficiently general it must have
both scalar and vector parts : the first gives one, and the second
three, scalar equations ; and these are required to determine com-
pletely the four scalar elements of the unknown quaternion.
176.] Thus Tq = a
being but one scalar equation, gives
q = aJJr,
where r is any quaternion whatever.
Similarly Sq — a
gives q — a +6,
where d is any vector whatever. In each of these eases, only one
scalar condition being given, the solution contains three scalar in-
determinates. A similar remark applies to the following :
Trq = a
gives q = x + ad,
and SUq = cos a,
gives q = 006'^ ,
in each of which x is any scalar, and any unit vector.
177.] Again, the reader may easily prove that
r.aVq^p,
^'^ QUATEENIONS. [178.
where a is a given vector, gives, by putting Sq = x,
Faq = p + cca.
Hence, assuming Saq = y,
we have aq=y + iDa + p,
or ? = «+yo-i + a-^j8.
Hercj the given equation being equivalent to two scalar con-
ditions, the solution contains two scalar indeterminates.
178.] Next take the equation
Faq = p.
Operating by 8.a-\ we get
Sq = 8a-^fi,
so that the given equation becomes
ra{Sa-^p+rq) = p,
or FaFq = ^-aSa-^fi = aVa'^
From this, by § 168, we see that
rq = a-^{x + aVa-^fi),
whence q = Sa-i/3 + a-^ {« + a Fa-i/S)
= a-i(/3 + a!),
and, the given equation being equivalent to three scalar conditions,
but one undetermined scalar remains in the value of q.
This solution might have been obtained at once, since our equation
gives merely the vector of the quaternion aq, and leaves its scalar
undetermined.
Hence, taking a; for the scalar, we have
aq = Saq + Vaq
179.] Finally, of course^ from
0^ = 13,
which is equivalent to four scalar equations, we obtain a definite
value of the unknown quaternion in the form
q = a-i^.
180.] Before taking leave of linear equations, we may mention
that Hamilton has shewn how to solve any linear equation con-
taining an unknown quaternion, by a process analogous to that
which he employed to determine an unknown vector from a linear
and vector equation j and to which a large part of this Chapter has
been devoted. Besides the increased complexity, the peculiar fea-
ture disclosed by this beautiful discovery is that the symbolic
equation for a linear quaternion function, corresponding to the cubic
1 8 3- J SOLUTION OF EQUATIONS. 97
in (^ of § 162, is a biquadratic, so that the inverse function is .given
in terms of the first, second, and third powers of the direct function.
In an elementary work like the present the discussion of such a
question would be out of place : although it is not very difficult to
derive the more general result by an application of processes already
explained. But it forms a curious example of the well-known fact
that a biquadratic equation depends for its solution upon a cubic.
The reader is therefore referred to the Mements of (Quaternions,
p. 491.
181.] The solution of the following frequently-occurring par-
ticular form of linear quaternion equation
aq + qb = c,
where a, b, and c are any given quaternions, has been effected by
Hamilton by an ingenious process, which was applied in § 133 (5)
above to a simple case.
Multiply the whole bi/ Ka, and into b, and we have
T'^a.q + Ka.qb=Ka.c,
and aqb-\-qb'^ = cb.
Adding, we have
q {T^a + b'^ + 2Sa.b) = Ka.c + cb,
from which q is at once found.
To this form any equation such as
a'qh'+c'qd' = e'
can of course be reduced, by multiplication by c'-^ and into b'"'^.
183.] As another example^ let us find the differential of the cube
root of a quaternion. If ^3 _ ,.
we have q'^dq + qdq.q + dq.q^ = dr.
Multiply by q, and into q~^, simultaneously, and we obtain
q^dq.q~^ + q^dq + qdq.q = qdr.q-^.
Subtracting this from the preceding equation we have
dq.q^—q^dq.q~^= dr—qdr.q~^,
or dq.q^—q^dq = dr.q—qdr,
from which dq, or d{r^), can be found by the process of last section.
The method here employed can be easily applied to find the
differential of any root of a quaternion.
183.] To shew some of the characteristic peculiarities in the
solution even of quaternion equations of the first degree when they
are not sufficiently general, let us take the very simple one
aq = qb,
and give every step of the solution, as practice in transformations.
H
y** QUATERNIONS. [183.
Apply Hamilton's process (§181), and we get
T^a.q = Ka.gh,
qh^ = aqb.
These give q(THJrb'^-2bSa,) = 0,
so that the equation gives no real finite value for q unless
or b = Sa + l3TFa,
where /3 is some unit-vector.
By a similar process we may evidently shew that
a = 8b + aTVb,
a being another unit-vector.
But, by the given equation,
Ta = Tb,
or S^a + T^ra = SH + TWb;
from which, and the above values of «• and b, we sec that we may
write So, Sb
Wa = Wb=^' '^PP°''-
If, then, we separate q into its scalar and vector parts, thus
q = r + p,
the given equation becomes
{a. + a){r + p) = (r + p)(a + ^) (1)
Multiplying out we have
r{a—l3) = pfi — ap,
which gives iS{a—p)p = 0,
and therefore p = Fy{a—fi),
where y is an undetermined vector.
We have now
r{a—p,) = p^-ap
= ry{a-^).l3-aFy{a-p)
= y{Safi+l)-{a-^)Spy + y{l+Sal3)-{a-fi)Say
= -ia-l3)Sia + fi)y.
Having thus determined r, we have
q=-S{a + p)y+Fyia-p)
2q=-{a + p)y-y{a + fi) + y{a-p)-ia-p)y
= —2ay—2y^.
Here, of course, we may change the sign of y, and write the solution
of aq = qb
in the form q = ay + yfi,
where y is any vector, and
a = UFa, /3 = UFb.
185.] SOLUTION OF EQUATIONS. 99
To verify this solution, we' see by ( 1 ) that we require only to
shew that aq =. qB.
But their common value is evidently
— y + ay/3.
It will be excellent practice for the student to represent the terms
of this equation by versor-arcsj as in § 54, and to deduce the above
solution from the diagram directly. He will find that the solution
may thus be obtained almost intuitively.
184.J No general method of solving quaternion equations of the
second or higher degrees has yet been found ; in fact, as will be
shewn immediatelyj even those of the second degree involve (in
their most general form) algebraic equations of the sixteenth degree.
Hence, in the few remaining sections of this Chapter we shall con-
fine ourselves to one or two of the simple forms for the treatment
of which a definite process has been devised. But firsts let us
consider how many roots an equation of the second degree in an
unknown quaternion must generally have.
If we substitute for the quaternion the expression
w-\rix-vjy + hz (§80),
and treat the quaternion constants in the same way, we shall have
(§ 80) four equations, generally of the second degree, to determine
w, X, y, z. The number of roots will therefore be 2* or 16. And
similar reasoning shews us that a quaternion equation of the mth
degree has w* roots. It is easy to see, however, from some of the
simple examples given above (§§ 175-178, &c.) that, unless the
given equation is equivalent to four scalar equations, the roots will
contain one or more indeterminate quantities.
185.] Hamilton has effected in a simple way the solution of the
quadratic ^^ = qa-{- h,
or the foUowingj which is virtually the same (as we see by taking
the conjugate of each side),
(f = aq + h.
He puts q—\{a + w + p),
where w is a scalar, and p a vector.
Substituting this value in the first equation, we get
a^ ^{iv + pf + 2wa + ap + pa = 2 {a^ -irWa-\- pa) + ^h,
or (M; + /3)^ + i2p— pa = a^ + 4^.
If we put Fa = a, S (a^ + 4b) = e, V{a^ + 45) = 2 y, this becomes
{w + pY + 2Vap = c+2y;
H a
100 QUATERNIONS. [l86.
which, by equating separately the scalar and vector parts, may be
broken up into the two equations
26)2 + p2 =: e,
V[w-\-ci)p = y.
The latter of these can be solved for p by the process of § 156, or
more simply by operating at once by S.a which gives the value of
S{w + a)p. If we substitute the resulting value of p in the former
we obtain, as the reader may easily prove, the equation
{w^-a^) (w*_cK>2 + y2)_(ray)2 = 0.
The solution of this scalar cubic gives six values of w, for each of
which we find a value of p, and thence a value of q.
Hamilton shews {Lectures, p. 633) that only two of these values
are real quaternions, the remaining four being biquaternions, and
the other ten roots of the given equation being infinite.
Hamilton farther remarks that the above process leads, as the
reader may easily see, to the solution of the two simultaneous
equations q + r = a,
qr = -b;
and he connects it also with the evaluation of certain continued
fractions with quaternion constituents. (See the Miscellaneous Ex-
amples at the end of the volume.)
186.] The equation q^ = aq+qb,
though apparently of the second degree, is easily reduced to the
first degree by multiplying 6y, and into, q~^, when it becomes
l=q-^a + bq-'^,
and may be treated by the process of § 181.
187.] The equation f' = aqb,
where a and h are given quaternions, gives
q{aqb) = {a,qb)q;
and, by § 54, it is evident that the planes of q and aqh must coincide,
A little consideration will shew that the solution depends upon
drawing two arcs which shall intercept given arcs upon each of two
great circles ; while one of them bisects the other, and is divided by
it in the proportion oi m: 1.
EXAMPLES TO CHAPTER V. 101
EXAMPLES TO CHAPTER V.
1. Solve the following equations: —
(a.) V.apP = V.ay^.
{h.) apfip = papj3.
(c.) ap + pP = y.
(d.) S.a^p + ^Sap — aVfip = y.
(e.) p + ap^= afi.
(/.) ap^p = p^pa.
Do any of these impose any restriction on the generality of a and j3 ?
2. Suppose p = ix+Jy + iz,
and (j)p = aiSip + hjSjp + ckSJcp ;
put into Cartesian coordinates the following equations : —
{a.) T4>p=l.
(b.) Spil>^p=-l.
(c.) S.p{<t>^-p^)-^p = -l.
{d.) Tp = T4Up.
3. If X, p,, V be any three non-coplanar vectors, and
q = F/xi'.(/)\+ FvX.(j)ix+ V\p,.(\>v,
shew that q is necessarily divisible by S.\p,v.
Also shew that the quotient is
^2-2 6,
where Vep is the non-commutative part of <^p.
Hamilton, Elements, p. 442.
4. Solve the simultaneous equations : —
Sap =0,1
^■> S.ap<bp = O.S
S.ap<pp
Sap =0,7
^"•^ Sp<l>p =0.5
Sap =0,1
^ ■' S.aipxp = 0. )
5. If # = S/3/Sap+ Frp,
where r is a given quaternion, shew that
I = S {8.ay,a^asS.fisfi^^^) + ^S{r Fa^a^ . r/S^^i) + SrlS.apr - 2 (/Sar/S/Sr) + SrTr^
and m4>-^<T='2{ra^a^S.^^0^<T) + l.r.ar{r^<T.r)+ VarSr- VrStrr.
Lectures, p. 561.
102 QUATERNIONS.
6. If [jog'] denote J>q~V''
{pqr) „ . S.plqr],
to''] » {pqr) + lrq']Sp + lpr]Sq + \jip]Sr,
and (i'?''*) !! '^'F L?***] >
shew that the following relations exist among any five quaternions
=jo{qrst) + q{rsip) + r{stpq) + s{tpqr) + t{pqrs),
and q{prs() =-[rsf\Spq—[stp~\Srq + [tpr'\Ssq—[prs]Stq.
Elements, p. 492.
7. Shew that if t^, i|f be any linear and vector functions, and
a, /3, y rectangular unit-vectors, the vector
e = V{4>a\lfa + ^fif^ + (j>r^y)
is an invariant. [This will be immediately seen if we write it in
the form 6 = F.<^V^p,
which is independent of the directions of a, )3, y. But it is good
practice to dispense with V.]
If # = S^i^Cft
and y^rp = ^rjiSC-yp,
shew that this invariant may be expressed as
-Sr#C or 2F7ji(^fi.
Shew also that cfi-ijfp—yjfcpp = F6p.
The scalar of the same quaternion is also an invariant, and may be
written as —'S2^Sr]r]j^SC(i
8. Shew that if <^p = aSap + ^Sfip + ySyp,
where a, ^, y are any three vectors, then
-<t>-^pS^.afiy = aiSa,p + ^,S^^ + yj,SY^p,
where a^ = Vfiy, &c.
9. Shew that any self-conjugate linear and vector function may
in general be expressed in terms of two given ones, the expression
involving terms of the second order.
Shew also that we may write
(^ +2; = a (OT-|-a;)2 + 5 (ct + a;) (oj +y) + c(<B +^)2,
where a, I, c, x, y, z are scalars, and ct and co the two given func-
tions. What character of generality is necessary in tn- and w ? How
is the solution affected by non-self-conjugation in one or both ?
10. Solve the equations : —
(a) q^ = Zqi+lOj.
{b.) q^ =2q + i.
(c.) qaq = bq + c.
(d.) aq = qr = rb.
EXAMPLES TO CHAPTER V. 103
11. Shew that ^FVcjyp = mVV(j)-'^p.
12. If (^ be self-conjugate, and a, /3, y a rectangular system,
S.Fa(f>ari3(})^Fy(f>y = 0.
13. (f)\l/ and yj/cp give the same values of the invariants m, m^,m,^.
14. If <^' be conjugate to <^, <^<^' is self-conjugate.
1 5. Shew that ( Va&f + ( Y^fff + ( Yye)^ = 26^
if a, /3, y be rectangular unit-vectors.
16. Prove that V^ {<j)—ff)p = —pV^g+2,Vg.
17. Solve the equations : —
'{a.) <^2 _ ^ .
{b.) ^ + x = i^, I
where one, or two, unknown linear and vector functions are given
in terms of known ones. (Tait, Proc. JR. S. JE- 1870-71.)
18. If <^ be a self-conjugate linear and vector function, £ and rj
two vectors, the two following equations are consequences one of
the other, viz. : — f _ F.Tj^rj
V ^ rm
Si.rj<pri4>^ri /S^.f^^^^^'
From either of them we obtain the equation
This, taken along with one of the others, gives a singular theorem
when translated into ordinary algebra. What property does it give
of the surface S.p(fip(j)^p = 1 ?
CHAPTER VI.
GEOMETRY OP THE STRAIGHT LINE AND PLANE.
188.] Having, in the five preceding Chapters, given a brief
exposition of the theory and properties of quaternions, we intend
to devote the rest of the work to examples of their practical appli-
cation, commencing, of course, with the simplest curve and surface,
the straight line and the plane. In this and the remaining Chapters
of the work a few of the earlier examples will be wrought out in
their fullest detail, with a reference to the first five whenever a
transformation occurs ; butj as each Chapter proceeds, superfluous
steps will be gradually omitted, until in the later examples the full
value of the quaternion processes is exhibited.
189.] Before proceeding to the proper business of the Chapter we
make a digression in order to give a few instances of applications
to ordinary plane geometry. These th-e student may multiply in-
definitely with great ease.
(a.) Euclid, I. 5. Let a and ^ be the vector sides of an iso-
sceles triangle ; /3— a is the base, and
Ta = T/3.
The proposition will evidently be proved if we shew that
a(a-^)-i=X/3(/3-a)-i (§ 52).
This gives a(a-^)-i= (/3— a)-i/3,
or (^— a)a = /3(a— j3),
or _a2 = -/32.
(5.) Euclid, I. 32. Let ABC be the triangle, and let
u-= = r»
AB
189.] GEOMETET OF STRAIGHT LINE AND PLANE. 105
where y is a unit-vector perpendicular to the plane of the triangle.
If ^ = 1, the angle GAB is a right angle (§ 74). Hence
4=^^(§74). Let^ = m^, C=n'l- We have
UlG=y'UAB,
UCB = y''UCA,
UBA=y"'UBC.
Hence UBA=y'^.y''.y^UAB,
or 1 = y+™+»>.
That is l-\-m + n =2,
or A + B+C=ii.
This is, properly speaking, Legendre's proof ; and might have been
given in a far shorter form than that above. In fact we have for
any three vectors whatever,
/3y a
which contains Euclid's proposition as a mere particular case.
(c.) Euclid, I. 35. Let y3 be the common vector-base of the
parallelograms, a the conterminous vector-side of any
one of them. For any other the vector-side is a + a?/3
(§ 28), and the proposition appears as
Tn{a + xp) = TV^a (§§ 96, 98),
which is obviously true.
{d.) In the base of a triangle find the point from which lines,
drawn parallel to the sides and limited by them, are
equal.
If a, j3 be the sides, any point in the base has the vector
p = (1— ar)a+a;/3.
For the required point
which determines x.
Hence the point lies on the line
which bisects the vertical angle of the triangle.
This is not the only solution, for we should have written
T(l-a))Ta = Ti»!Tp,
instead of the less general form above wMck tacitly assumes that 1—x
and cc are positive. We leave this to the student.
106 QUATERNIONS. [iQO.
(e.) If perpendiculars be erected outwards at the middle points
of tlie sides of a triangle^ each being proportional to
the corresponding sidcj the mean point of the triangle
formed by their extremities coincides with that of the
original triangle. Find the ratio of each perpendicular
to half the corresponding side of the old triangle that
the new triangle may be equilateral.
Let 2a, 2 /3j and 2 (a + y3) be the vector-sides of the triangle, i a
unit-vector perpendicular to its plane, e the ratio in question. The
vectors of the corners of the new triangle are (taking the corner
opposite to 2/3 as origin)
/Oj = a + eia,
P2 = 2a + /3 + ei/3,
P3 = a + /3— ei((a + /3).
From these
*(ft + P2 + /'3) = H4a+2;8) = k (2a-|-2(o + /3)),
which proves the first part of the proposition.
For the second part, we must have
^fe— Pi) = ^(P3— Pa) = 2'(pi— Pa).
Substituting, expanding, and erasing terms common to all, the
student wUl easily find 3 gS _ j _
Hence, if equilateral triangles be described on the sides of any tri-
angle, their mean points form an equilateral triangle.
190.] Such applications of quaternions as those just made are of
course legitimate, but they are not always profitable. In fact, when
applied to plane problems, quaternions often degenerate into mere
scalars, and become (§33) Cartesian coordinates of some kind, so
that nothing is gained (though nothing is lost) by their use. Before
leaving this class of questions we take, as an additional example, the
investigation of some properties of the ellipse.
191.] We have already seen (§31 {h)) that the equation
p = acos5 + /3sinfl
represents an ellipse, Q being a scalar which may have any value.
Hence, for the vector-tangent at the extremity of p we have
Ap • ^ ^ „
OT = -^ = — asmd + i3cos0,
do
It
which is easily seen to be the value of p when 6 is increased by - •
Thus it appears that any two values of p, for which difiers by
1 94-] GEOMETRY OP STRAIGHT LINE AND PLANE. 107
IT'
- , are conjugate diameters. The area of the parallelogram circum-
scribed to the ellipse and touching it at the extremities of these
diameters is, therefore, by § 96,
^TFp-^ = 4yr(acos0 + /3sin9)(— asine + /3eose)
= 4yFa/3,
a constant, as is well known.
193.] For equal conjugate diameters we must have
y(aeose + /3sin0) = y(— a sin 9 4-/3 cos 0),
or (a^— /3^) (cos^^— sin20) + 4^a/3cosesini9 = 0,
a^ — B^
or tan 2 9 = „ „ '^ •
The square of the common length of these diameters is of course
a2 + ^^
,
2
because we see at once from § 191 that the sum of the squares of
conjugate diameters is constant.
193.] The maximum or minimum of p is thus found ;
dTp 1_ dp_
de ~~T^''dd'
= — -^{ — (a^— 13^) cose sine + Sap icos^e—sm^0)).
For a maximum or minimum this must vanish *, hence
tan 2^= -5 — ^„,
a^ — ^^
and therefore the longest and shortest diameters are equally inclined
to each of the equal conjugate diameters. Hence, also, they are at
right angles to each other.
194.] Suppose for a moment a and ;3 to be the greatest and least
semidiameters. Then the equations of any two tangent-lines are
p = a cos ^ + ;8 sin 5 + «(— a sin ^ + /3 cos 6),
p = a cos ^1 H- j3 sin 0^ + Xy(^—a sin ^j -)- /3 cos 0-^.
If these tangent-lines be at right angles to each other
<?(— asin(9-|-/3cosfl)(— asin^i + /3cosei) = 0,
or o? sin 6 sin 6^ -)- /3^ cos 5 cos ^j = 0.
dB
A little reflection will shew him that the latter equation involves an absurdity.
The student must carefully notice that here we put -j— = 0, and not ^ = 0.
civ
108 QUATEEJSriOKS. [195.
Also, for their point of intersection we have, by comparing coeffi-
cients of a, /3 in the above values of p,
cos 6—xsmd = cos 6^ —x^ sin 6^ ,
sin O + x cos 6 = sin 6^ + x^ cos d-y .
Determining x-y from these equations, we easily find
the equation of a circle ; if we take account of the above relation
between 6 and d^.
Also, as the equations above give x = — x^, the tangents are equal
multiples of the diameters parallel to them ; so that the line joining
the points of contact is parallel to that joining the extremities of
these diameters.
195.] Finally, when the tangents
p = acosd +y3 sin5 +x (— asinfl +;Scos0),
p = a cos 0^ + j8 sin d^ + x^ (—a sin 6^ + ^ cos 0^),
meet in a given point p = aa + bp,
we have a = cos 6 — x sin = cos 6^ — x^ sin 0^,
h = sin 0-\-x cos = sin 0^ + x^ cos 0-^ .
Hence x"^ = a^ + b^—1 = xl
and a cos + b sin = 1 = a cos ^j + J sin 0^
determine the values of and x for the directions and lengths of
the two tangents. The equation of the chord of contact is
p = y{a cos 6 + /3 sin 6) + (1 —y) (a cos ^^ + /3 sin 0^).
If this pass through the point
p=jia + q^,
we have ^ = ycos0 + (l— j^)cos9i,
q = 2/svD.0 + {\ —y) sin 0^,
from which, by the equations which determine and 5, , we get
a])-irl(i=yJr\—y= 1.
Thus if either a and h, or ^ and ^, be given, a linear relation con-
nects the others. This, by § 30, gives all the ordinary properties of
poles and polars.
196.] Although, in §§ 28-30, we have already g^ven some of the
equations of the line and plane, these were adduced merely for their
applications to anharmonic coordinates and transversals ; and not
for investigations of a higher order. Now that we are prepared to
determine the lengths and inclinations of lines we may investigate
these and other similar forms anew.
200.] GEOMETRY OF STRAIGHT LINE AND PLANE. 109
197.] The equation of the indefinite line d/rawn through the origin
0, of which the vector OA, = a, forms apart, is evidently
p = soa,
or p II a,
or Fap = 0,
or Up =: Ua;
the essential characteristic of these equations being that they are
linear, and involve one indeterminate scalar in the value of p.
We may put this perhaps more clearly if we take any two
vectors, /3, y, which, along with a, form a non-coplanar system.
Operating with S.Va^ and S.Vay upon any of the preceding equa-
tions, we get S.afip = 0,1
and S.ayp = Q.\ *■ ''
Separately, these are the equations of the planes containing a, /3,
and o, y ; together, of course, they denote the line of intersection.
198.] Conversely, to solve equations (1), or to find p in terms of
known quantities, we see that they may be written
-S.pFa/3 = 0,-)
S.pFay = 0,)
so that p is perpendicular to Fa^ and Fay, and is therefore parallel
to the vector of their product. That is,
pII KFajSray,
II -aS.a^y,
or p = xa.
199.] By putting p— ;3 for p we change the origin to a point S
where 0£ = — ^, or ^0 = y3 ; so that the equation of a line parallel
to a, and passing through the extremity of a vector /3 drawn from
the origin, is p—^ = xa,
or p = p + xa.
Of course any two parallel lines may be represented as
p = /3 +xa,
p = pj^+Xj^a;
or Fa{p-fi) = 0,
Fa{p-I3,) = 0.
200.] The equation of a line, drawn through the extremity of ^, and
meeting a perpendicularly, is thus found. Suppose it to be parallel
to y, its equation is p = ^ + xy.
To determine y we know, first, that it is perpendicular to o, which
gives Say = 0-
110 QUATERNIONS. [2OI.
Secondly, o, ^, and y are in one plane, which gives
S.a^y - 0.
These two equations give y |{ r.aFaj3,
whence we have p =: j3 + soa Vafi.
This might have been obtained in many other ways ; for instance,
we see at once that
/3 = a- la/3 = a-^Safi + a-Wa^.
This shews that ar^Va^ (which is evidently perpendicular to a)
is coplanar with a and /3, and is therefore the direction of the re-
quired line ; so that its equation is
p = fi+ya-WaP,
V
the same as before if we put — ^-5- for x.
la
201.J By means of the last investigation we see that
—arWa^
is the vector perpendicular drawn from the extremity of /3 to the
line p = xa.
Changing the origin, we see that
-a-ira(j3->/)
is the vector perpendicular from the extremity of /3 upon the line
p = y + xa.
203.] The vector joining £ (where OJS = fi) with any point in
p =■ y + xa
is y + Xa—p.
Its length is least when
dT{y+xa—0) = 0,
or Sa{y + xa—^) = 0,
i. e. when it is perpendicular to a.
The last equation gives
■xa^+Sa{y-^) = 0,
or xa=—a'' ^Sa{y — /3) .
Hence the vector perpendicular is
y-^-a-^Sa{y-0),
or a-^Fa{y-fi)=-a-^Fa{l3—y),
which agrees with the result of last section,
203.] To find the shortest vector distance between two lines
p = fi + xa,
and Pi=/?i + «'iai;
204.J GEOMETRY OF STRAIGHT LINE AND PLANE. Ill
we must put dT{p—p^ =*0,
or S{p-p^){dp-dp{) = 0,
or S{p—pj){adx—aidxi) = 0.
Since x and x^ are independent^ this breaks up into the two eon-
'litioiis Sa{p-p,)=0,
Sajip-pj) = 0;
proving the well-known truth that the required line is perpendicular
to each of the given lines.
Hence it is parallel to Faa-^, and therefore we have
p—pi—l3 + xa—l3-^—x^ai = yFaai (1)
Operate by S.aaj and we get
This determines y, and the shortest distance required is
[_N'ote. In the two last expressions T before S is inserted simply
to ensure that the length be positive. If
/S'.aai(/3 — /3j) be negative,
then (§89) xS'.a^a(/3— ySj) is positive.
If we omit the T, we must use in the text that one of these two ex-
pressions which is positive.J
To find the extremities of this shortest distance, we must operate
on (1) with S.a and S.a^. We thus obtain two equations, which
determine x and x^, as y is already known.
A somewhat different mode of treating this problem will be dis-
cussed presently.
204.] In a given- tetrahedron to find a set of rectangular coordinate
axes, such that each axis shall ^ass through a pair of opposite edges.
Let a, /3, y be three (vector) edges of the tetrahedron, one corner
being the origin. Let p be the vector of the origin of the sought
rectangular system, which may be called i, j, k (unknown vectors).
The condition that i, drawn from p, intersects a is
S.iap = (1)
That it intersects the opposite edge, whose equation is
7^ = ^ + x{^-y),
the condition is
S.i(fi-y){p-^)=0, or Si{{^-y)p-M = (^ (2)
There are two other equations hke (1), and two like (2), which can
be at once written down.
^^^ QUATERNIONS. [205.
Put p-y=a^, y-a = ^i, a-fi = y^,
r^y = a^, Fya = /a^, Ta/S = y^,
and the six become
S.iap = 0, S.ia^p —Sia^ = 0,
S.Jpp = 0, S.j0,p-8jp^ = 0,
S.kyp = 0, S.hy-^p-Shy^ = 0.
The two in i give i \\ aSa^-piSaa^ + Sarfi).
Similarly,
J\\^Sfi2p-p{Sfi^2 + SM> and i\\YSy2P-p{Syy^ + Sysp).
The conditions of rectangulaffity, viz.,
SiJ = 0, SJi = 0, SM = 0,
at once give three equations of the fourth order, the first of which is
= Safi Sa^p S^^p-Sap Sa^{Sfifi^ + Sj3^p)-Sfip Sp^p{Saa^ + Sa^p)
+ p2 [Saa^ + Sa^p) {S^p^ + Sj3^).
The required origin of the rectangular system is thus given as
the intersection of three surfaces of the fourth order.
205.] The equation Sap =
imposes on p the sole condition of being perpendicular to a ; and
therefore, being satisfied by the vector drawn from the origin to
any point in a plane through the origin and perpendicular to a, is
the equation of that plane.
To find this equation by a direct process similar to that usually
employed in coordinate geometry, we may remark that, by § 29, we
may write p = xj3 +yy,
where /3 and y are any two vectors perpendicular to a. In this
form the equation contains two indeterminates, and is often useful ;
but it is more usual to eliminate them, which may be done at once
by operating by Sm, when we obtain the equation first written.
It may also be written, by eliminating one of the indeterminates
only, as T^p = ya,
where the form of the equation shews that Sa^ = 0.
Similarly we see that
Sa (p-/3) =
represents a plane drawn through the extremity of ^ and perpen-
dicular to a. This, of course, may, hke the last, be put into various
equivalent forms.
306,] The line of intersection of the two planes
8.a (p-/3) = 0, )
and 5.ai(p-^i)=0,) ^ >
2o8.] GEOMETRY 0? STRAIGHT LINE AND FLAKE. 113
contains all points whose value of p satisfies both conditions. But
we may write (§ 92), since a, a^, and Faa-^ are not coplanar,
pS.aa-^Vaa-^^ — Vaa^SMa^p^ V.a-J^aai8ap+ F.F{aa^)aSa^p,
or, by the given equations,
—pT^ Vaa-^ = r.d^ Vaa^Sa^ + V. r{aa^ aSa^^ + x Yaa^, (2)
where x, a scalar indeterminate, is put for S.aa^p which may have
any value. In practice, however, the two definite given scalar
equations are generally more useful than the partially indeterminate
vector-form which we have derived from them.
When both planes pass through the origin we have /3 = /S^ = 0,
and obtain at once ^ ^ ^ jr^^
as the equation of the line of intersection.
207.] The plane passing through the origin, and through the line of
intersection of the two planes (1), is easily seen to have the equation
Sa^^iSap — SajSSaip = 0,
or S{aSa^l3-^—a-^SaP)p = 0.
For this is evidently the equation of a plane passing through the
origin. And^ if p he such that
Sap = Safi,
we also have Saj^p = Sa^^^,
which are equations (1).
Hence we see that the vector
aSa^^jSi — ajSaj3
is perpendicular to the vector-line of intersection (2) of the two
planes (1), and to every vector joining the origin with a point in
that line.
The student may verify these statements as an exercise.
208.] To find the vector-perpendicular from the extremity of ^ on
the plane Sap = 0,
we must note that it is necessarily parallel to a, and hence that the
value of p for its foot is p — ^^^a,
where xa is the vector-perpendicular in question.
Hence Sa {j3 + xa) = 0,
which gives xa^:= —Sa^,
or Xa = —a~^Sa/3.
Similarly the vector-perpendicular from the extremity of /3 on the
may easily be shewn to be
-a-'^Sa(l3-y).
I
114 QUATERNIONS. [209.
209.] The equation of the plane which passes through the ecctremities
of a,^,y may be thus found. If p be the vector of any point in it)
p—a, a—p, and /3— y lie in the plane, and therefore (§101)
S.{p-a){a-^){fi-y)=0,
or Sp{ra^-{- Vfiy+ rya)-S.a^y = 0.
Hence, if 6 = a; ( F"a/3 + T/Sy + Fya)
be the vector-perpendicular from the origin on the plane containing
the extremities of a, y3, y, we have
6 = (ra/3+ r/3y+ Fyay^S.a^y.
From this formula, whose interpretation is easy, many curious pro-
perties of a tetrahedron may be deduced by the reader. Thus, for
instance, if we take the tensor of each side, and remember the
result of § 100, we see that
T{ral3+rfiy+rya)
is twice the area of the base of the tetrahedron. This may he more
simply proved thus. The vector area of base is
ir{d-fi) (y-/3) =-iiral3+ r^y+ Fya).
Hence the sum of the vector areas of the faces of a tetrahedron,
and therefore of any solid whatever, is zero. This is the hydrostatic
proposition for solids immersed in a fluid subject to no external
forces.
310.] Taking any two lines whose equations are
p = 13 + xa,
p =: jSj^ + X^Oi,
we see that S.aaj(p — 6) ^
is the equation of a plane parallel to both. Which plane, of course,
depends on the value of 8.
Now if 8 = /3, the plane contains the first line ; if 8 = ^1, the
second.
Hence, liyVaa^ be the shortest vector distance between the lines,
we have 5_„„^ {fi-^^-y Faa^) = 0,
or TiyFaa^) = m(/3-^i) UFaa^,
the result of § 203.
211.J Find the equation of the plane, passing through the origin,
which makes equal angles with three given lines. Also find the angles
in question.
Let a, y3, y be unit-vectors in the directions of the lines, and let
the equation of the plane be
Sbp = 0.
2I3.J GEOMETRY OF STEAIGHT LINE AND PLANE. 115
Then we have evidently
Sab = /S/38 = Syb = x, suppose,
where ^
Tb
is the sine of each of the required angles.
But (§ 92) we have
bS.a/Sy = X iFa^+ F^y+ Fya).
Hence S.p ( VajS + V/3y + Fya) =
is the required equation ; and the required sine is
S.a^y
~ T{ra^+rpy+rya)'
312.] Find the locus of the middle points of a series of straight
lines, each parallel to a given plane and having its extremities*in two
fixed lines.
Let 8yp —
be the plane, and p = yg^a-^^ ^^ ^-y^x^a^,
the fixed lines. Also let x and x-^ correspond to the extremities of
one of the variable lines, is- being the vector of its middle point.
Then, obviously, 2-a! = ^ + xa-\-^^+x-^a^.
Also 8y{fi—^^->rXa—x^a^ = 0.
This gives a linear relation between so and x-^ , so that, if we sub-
stitute for Xj^ in the preceding equation, we obtain a result of the
form ^^8+a;e,
where 8 and e are known vectors. The required locus is, therefore,
a straight line.
313.] Three planes meet in a point, and through the line of inter-
section of each pai/r a plane is drawn perpendicular to the third ; prove
that, in general, these planes pass through the same line.
Let the point be taken as origin, and let the equations of the planes
^e Sap = 0, Sl3p = 0, Syp = 0.
The line of intersection of the first two is || FajB, and therefore the
normal to the first of the new planes is
F.yFajB.
Hence the equation of this plane is
S.pF.yFa^ = 0,
or SfipSay—SapS^y = 0,
and those of the other two planes may be easily formed from this
by cyclical permutation of a, /3, y.
I a
116 QUATERNIONS. [214.
We see at once that any two of these equations give the third by
addition or subtraction, which is the proof of the theorem.
214.] Griven any number of points A, B, G, 8fc., whose vectors
{from the origin) are a^, Og, a.^, 8fc.,find the plane through the origin
for which the sum of the squares of the perpendiculars let fall upon it
from these points is a maximum or minimum.
Let ^wp =
be the required equation, with the condition (evidently allowable)
IW= 1.
The perpendiculars are (§ 208) — ■nr~^*S'OTai, &c.
Hence ^S^-^a
is a maximum. This gives
"^.SisaSadiTt! = ;
and the condition that ■zn- is a unit-vector gives
SnydvT = 0.
Hence, as d-sr may have any of an infinite number of values, these
equations cannot be consistent unless
where a; is a scalar.
The values of o are known, so that if we put
^ is a given self-conjugate linear and vector function, and therefore
a; has three values {g^, g^, g^, § 164) which correspond to three
mutually perpendicular values of -57. For one of these there is a
maximum, for another a minimum, for the third a maximum-
minimum, in the most general case when g^, g.^., g^ are all different.
215.] The following beautiful problem is due to MaccuUagh.
Of a system of three rectangular vectors, passing through the origin,
two lie on given planes, find the locus of the third.
Let the rectangular vectors be ot, p, a. Then by the conditions
of the problem gsyp = Spa = Sa^ = 0,
and iSara- = 0, S^p = 0.
The solution depends on the elimination of p and ot among these
five equations. [This would, in general, be impossible, as p and ■in-
between them involve six unknown scalars ; but, as the tensors are
(by the very form of the equations) not involved, the five given
equations are necessary and suflicient to eliminate the four unknown
scalars which are really involved. Formally to complete the re-
quisite number of equations we might write
Ts^ = a, Tp = h,
but a and h may have any values whatever.]
EXAMPLES TO CHAPTER VI. 117
From Sasr = 0, /So-sr = 0,
we have in- = xFaa:
Similarly, from Sfip = 0, Sap = 0,
we have P = y Vj3a:
Substitute in the remaining equation
S'srp = 0,
and we have S.FaaF^a = 0,
or Sa<rSj3<T — cr^Sa^ = 0,
the required equation. As will be seen in next Chapter, this is a
cone of the second order whose circular sections are perpendicular
to a and /3. [The disappearance of x and y in the elimination in-
structively illustrates the note above.J
EXAMPLES TO CHAPTER VI.
1. What propositions of Euclid are proved by the mere /by»« of
the equation p = {l—ai)a + xj3,
which denotes the line joining any two points in space ?
2. Shew that the chord of contact, of tangents to a parabola
which meet at right angles, passes through a fixed point.
3. Prove the chief properties of the circle (as in Euclid, III) from
the equation p = a cos + ^ sin ;
where Ta = Tfi, and Sa^ = 0.
4. What, locus is represented by the equation
S^ap + p^= 0,
where Ta= I?
5. What is the condition that the lines
Fap = A Fa^P = ySi,
intersect? If this is not satisfied, what is the shortest distance
between them ?
6. Find the equation of the plane which contains the two parallel
lines ra(p-/3)=0, Fa{p-^i) = 0.
7. Find the equation of the plane which contains
ra(p-/3) = 0,
and is perpendicular to gyp — o.
8. Find the equation of a straight line passing through a given
point, and making a given angle with a given plane.
Hence form the general equation of a right cone.
118 QUATERNIONS.
9. What conditions must be satisfied with regard to a number of
given lines in space that it may be possible to draw through each
of them a plane in such a way that these planes may intersect in a
common line ?
10. Find the equation of the locus of a point the sum of the
squares of whose distances from a number of given planes is con-
stant.
11. Substitu^ "lines" for "planes" in (10).
12. Find the equation of the plane which bisects, at right angles,
the shortest distance between two given lines.
Find the locus of a point in this plane which is equidistant from
the given lines.
1 3. Find the conditions that the simultaneous equations
Sap = a, S^p = 6, Syp = c,
may represent a line, and not a point.
1 4. What is represented by the equations
{Sapf = {Sl3py^ = {Syp)^
where a, /3, y are any three vectors ?
15. Find the equation of the plane which passes through two
given points and makes a given angle with a given plane.
16. Find the area of the triangle whose corners have the vectors
a, /3, y.
Hence form the equation of a circular cylinder whose axis and
radius are given.
17. (Hamilton, Bishop Law's Fremium Ex., 1858).
{a.) Assign some of the transformations of the expression
/3— a'
where a and /3 are the vectors of two given points A and B.
{h.) The expression represents the vector y, or OC, of a point C
in the straight line AB.
(c.) Assign the position of this point C.
18. (Ibid.)
(a.) If a, /3, y, 8 be the vectors of four points. A, B, C, B, what
is the condition for those points being in one plane ?
(h.) When these four vectors from one origin do not thus ter-
minate upon one plane, what is the expression for the
volume of the pyramid, of which the four points are the
corners ?
(c). Express the perpendicular S let fall from the origin on
the plane ABC, in terms of a, y3, y.
EXAMPLES TO CHAPTER VI. 119
19. Find the locus of a point equidistant from the three planes
Sap = 0, S^p = 0, Syp = 0.
20. If three mutually perpendicular vectors be drawn from a
point to a plane, the sum of the reciprocals of the squares of their
lengths is independent of their directions.
21. Find the general form of the equation of a plane from the
condition (which is to be assumed as a definition) that any two
planes intersect in a single straight line.
22. Prove that the sum of the vector areas of the faces of any
polyhedron is zero.
CHAPTER VII.
THE SPHERE AND CYCLIC CONE.
216.] Aftee that of the plane the equations next in order of
simplicity are those of the sphere, and of the cone of the second
order. To these we devote a short Chapter as a valuable prepara-
tion for the study of surfaces of the second order in general.
217.] The equation y^ _ ^a
or p^ = (^,
denotes that the length of p is the same as that of a given vector a,
and therefore belongs to a sphere of radius Ta whose centre is the
origin. In § 107 several transformations of this equation were ob-
tained, some of which we will repeat here with their interpretations.
Thus ^(p + a)(p-a) =
shews that the chords drawn from any point on the sphere to the
extremities of a diameter (whose vectors are a and —a) are at right
angles to each other.
r(p + a)(p-a)= iTVap
shews that the rectangle under these chords is four times the area
of the triangle two of whose sides are a and p.
(0 = (p + a)"^a(/3 + a) (see § 105)
shews that the angle at the centre in any circle is double that at
the circumference standing on the same arc. All these are easy
consequences of the processes already explained for the interpretation
of quaternion expressions.
218.] If the centre of the sphere be at the extremity of a the
equation may be written
T{p-a) = Tp,
which is the most general form.
If Ta = T/3,
or a2 = /3^
2 2 O.J THE SPHERE AND CYCLIC CONE. 121
in which ease the origin is a point on the surfaee of .the sphere, this
becomes p^-2Sap = 0.
From this, in the form
Sp{p — 2a) =
another proof that the angle in a semicircle is a right angle is de-
rived at once.
219.] The converse problem is — Mnd the locus of tJiefeet of per-
pendiculars let fall from a given point (p=/3) on planes passing through
the origin.
Let Sap =
be one of the planes, then (§208) the vector-perpendicular is
— a-^Saj3,
and, for the locus of its foot,
p = /3 — a-i/S'a/3,
= orWap.
[This is an example of a peculiar form in which quaternions some-
times give us the equation of a surfaee. The equation is a vector
one, or equivalent to three scalar equations ; but it involves the
undetermined' vector a in such a way as to be equivalent to only-
two indeterminates (as the tensor of a is evidently not involved).
To put the equation in a more immediately interpretable form, a
must be eliminated, and the remarks just made shew this to be
possible.]
Now {p-^Y =a-Wap,
and (operating by S.fi)
S^p-fi^=-a-Wafi.
Adding these equations, we get
P^-S^P = 0,
so that, as is evident, the locus is the sphere of which y3 is a dia-
meter.
220.] To find the intersection of the two spheres
T(p-a) = h,
and ^(p-«i) = ^/3i.
square the equations, and subtract, and we have
2S{a-ai)p = a^-ai^-{^^-l3j^),
which is the equation of a plane, perpendicular to a— aj the vector
joining the centres of the spheres. This is always a real plane
whether the spheres intersect or not. It is, in fact, what is called
their Radical Plane.
122 QUATERNIONS. [221.
331.] Find the locus of a point the ratio of whose distances from
two given points is constant.
Let the given points be and A, the extremities of the vector a.
Also let P be the required point in any of its positions, and OP=p.
Then, at once, if n be the ratio of the lengths of the two lines,
T{p-a) = nTp.
This gives p^ — 2Sap + a^ = »2 p^,
or, by an easy transformation,
Thus the locus is a sphere whose radius is Tf- ^^> and whose
centre is at JB, where 0£ = 5- > a definite point in the line OA.
1—n^ ^
632.] ^in any line, OP, drawn from the origin to a given plane,
OQ be taken such that OQ.OP is constant, fnd the locus of Q.
Let Sap = a
be the equation of the plane, ct a vector of the required surface.
Then, by the conditions,
T'HT Tp = constant = 5^ (suppose),
and Z7«r = Up.
From these p = -s= — = 5- •
Substituting in the equation of the plane, we have
aw^ + b^Saw = 0,
which shews that the locus is a sphere, the origin being situated on
it at the point farthest from the given plane.
333.] FiMd the locus of points the sum of the squares of whose dis-
tances from a set of given points is a constant quantity. Find also the
least value of this constant, and the corresponding locus.
Let the vectors from the origin to the given points be oj, Oj,
a„, and to the sought point p, then
-c2 = {p-c^f+[p-a^f + + (p-a„)^
= np^-2Sp'2a+-S,{a^).
Otherwise (,_^«/= _ flilli^!! + (?#,
\ n' n n^
the equation of a sphere the vector of whose centre is — > i.e.
whose centre is the mean of the system of given points.
Suppose the origin to be placed at the mean point, the equation
becomes /.2 j. y („i\
p2 ^ _^__+±S5l1 (for 2a = 0, § 31 (e)).
2 26. J THE SPHERE AND CYCLIC CONE. 123
The right-hand side is negative, and therefore the equation denotes
a real surface, if ^2 ^ 2Ta^
as might have been expected. When these quantities are equal,
the locus becomes a point, viz. the new origin, or the mean point of
the system.
334.J If we differentiate the equation
Tp = Ta
we get Spdp — 0.
Hence {^ \i7), p is normal ^ the surface at its extremity, a well-
known property of the sphere.
If tn- be any point in the plane which touches the sphere at the
extremity of p, ta-— p is a line in the tangent plane, and therefore
perpendicular to p. So that
8p{'7!-p) = 0,
or S-arp = — Tp^ = a^
is the equation of the tangent plane.
225 .J If this plane pass through a given point B, whose vector
is fi, we have ^^^ ^ „2.
This is the equation of a plane, perpendicular to /3, and cutting
from it a portion whose length is
Tp '
If this plane pass through a fixed point whose vector is y we must
have spy = a^
so that the locus of /8 is a plane. These results contain all the
ordinary properties of poles and polars with regard to a sphere.
226.] A line drawn parallel to y, from the extremity of /3, has
the equation p — a^^
This meets the sphere p2 _ ^2
in points for which w has the values given by the equation
P^ + 2xSl3y-^x^y'^ = a^.
The values of a; are imaginary, that is, there is no intersection, if
The values are equal, or the line touches the sphere, if
aV+^^/3y = 0,
or S^l3y = y^P^-a^).
This is the equation of a cone similar and similarly situated to the
cone of tangent-lines drawn to the sphere, but its vertex is at the
centre. That the equation represents a cone is obvious from the
124 QUATERNIONS. [227,
fact that it is homogeneous in Ty, i.e. that it is independent of the
length of the vector y.
[It may be remarked that from the form of the above equation
we see that, if x and x' be its roots, we have
{xTy){x"I>y)=c?-fi\
which is Euclid, III, 35, 36, extended to a sphere.]
227.] Find the locus of the foot of the perpendicular let fall from
a given point of a sphere on any tangent-plane.
Taking the centre as origin, the equation of any tangent-plane
may be written ^^p ^ „2_
The perpendicular must be parallel to p, so that, if we suppose it
drawn from the extremity of a (which is a point on the sphere) we
have as one value of ■or
■cT = a-\-xp.
From these equations, with the help of that of the sphere
we must eliminate p and x.
We have by operating on the vector equation by S.'^
■sr^ = SaiiT+xS'STp
■=■ /iSatsr + ara^.
__ CT — a a^ (■or — a)
Hence p = = — 5 — 5
Taking the tensors, we have
(i!r2_^a^)2 = a2(ti^-a)^
the required equation. It may be put in the form
and the interpretation of this gives at once a characteristic property
of the surface formed by the rotation of the Cardioid about its axis
of symmetry.
228.] We have seen that a sphere, referred to any point what-
ever as origin, has the equation
T{p-a) = T^.
Hence, to find the rectangle under the segments of a chord drawn
through any point, we may put
p=xy;
where y is any unit-vector whatever. This gives
x^y^-2xSay+a^ = ^^,
and the product of the two values of x is
y
2 31.] THE SPHERE AND CYCLIC CONE. 125
.... • •
This is positive, or the vector-chords are drawn in the same direc-
tion, if T&<Ta,
i.e. if the origin is outside the sphere.
229.] A, B are fixed points s and, leing the origin and P a point
m space, jjp2 ^ ^pa = Qpa .
find the locus ofP, and explain the result when LAOB is a right, or
an obtuse, angle.
Let OJ = a, 0B = ^,6P=p, then
or p2_2^(a + y3)p=_(a2+/32),
or y{p-(a4-/3)}=^/(-2&/3).
While Sa^ is negative, that is, while LAOB is acute, the locus is a
sphere whose centre has the vector o + /3. If ASa/3=0, or LAOB=-,
the locus is reduced to the point
p = a + /3.
"If LAOB>- there is no point which satisfies the conditions.
230.] Bescriie a sphere, with its centre in a given line, so as to
pass through a given point and touch a given plane.
Let xa, where « is an undetermined scalar, be the vector of the
centre, r the radius of the sphere^ /3 the vector of the given point,
and Syp = a
the equation of the given plane.
The vector-perpendicular from the point xa on the given plane is
(§208) {a-xSya)y-''.
Hence, to determine x we have the equation
T.{a-x8ya)y-'^ = T{xa-^) = r,
so that there are, in general, two solutions. It will be a good
exercise for the student to find from this equation the condition
that there may be no solutioQj or two coincident ones.
231.] Bescribe a sphere whose centre is in a given line, and which
passes through two given points.
Let the vector of the centre be xa, as in last section, and let the
vectors of the points be ^ and y. Then, at once,
T{y-xa) =T{fi-xa) = r.
Here there is but one sphere, except in the particular case when we
have Ty = T^, and Say = Sa^,
in which case there is an infinite number.
126 QUATERNIONS. [232.
•'
The student should carefiiUy compare the results of this section
and the last, so as to discover why in general two solutions are
possible in the one case, and only one in the other.
232.] A sphere touches each of two straight lines, which do not
meet -. find, the locus of its centre.
We may take the origin at the middle point of the shortest dis-
tance (§203) between the given lines, and their equations will then
be p = a-\-x^,
where" we have, of course,
Sa^ = 0, xSa/3i = 0.
Let <r be the vector of the centre, p that of any point, of one of the
spheres, and r its radius ; its equation is
T{p-a) = r.
Since the two given lines are tangents, the following equations in x
and Xi must have pairs of equal roots,
2'(a4-«/8 — (7) = r,
T{-a + a;^Pi-a-)=zr.
The equality of the roots in each gives us the conditions
S^I3<T =/32((a-(r)2+»-2),
-S2/3i<T=^f((a + cr)2+r2).
Eliminating r we obtain
^-^S^fia-fil^S^fi^a- = (a-o-)2-(a + <r)2 =-45a(7,
which is the equation of the required locus.
[As we have not, so far, entered on the consideration of the qua-
ternion form of the equations of the various surfaces of the second
order, we may translate this into Cartesian coordinates to find its
meaning. If we take coordinate axes of so, y, z respectively parallel
to |3, /3i, a, it becomes at once
{x-\-myf^{jl-\-mxf' =^ pz,
where m and p are constants ; and shews that the locus is a hy-
perbolic paraboloid. Such transformations, which are exceedingly
simple in all cases, will be of frequent use to the student who is
proficient in Cartesian geometry, in the early stages of his study of
quaternions. As he acquires a practical knowledge of the new
calculus, the need of such assistance will gradually cease to be
felt.J
Simple as the above solution is, quaternions enable us to give one
vastly simpler. For the problem may be thus stated — Find the
locus of the point whose distances from two given, lines are equal.
2 34-] THE SPHEKE AND CYCLIC CONE. 127
And, with, the above notation, the equality of the perpendiculars is
expressed (§ 201) by
TV. (a -a)U^ = TV. (a + <t) U^^ ,
which is easily seen to be equivalent to the equation obtained above.
233.] Two spheres being given, shew that spheres which cut them at
given atigles cut at right angles another fixed sphere.
If be the distance between the centres of two spheres whose radii
are a and i, the cosine of the angle of intersection is evidently
¥ab
Hence, if a, a^, and p be the vectors of the centres, and «,«!,»• the
radii, of the two fixed, and of one of the variable, spheres ; A and
^1 the angles of intersection, we have
{p — af+a'^-\-r^= 2ar cos A,
{p—aj)^ +al+r^ = 2ajrcosAj^.
Eliminating the first power of r, we evidently must obtain a result
sueh as (p— /S)^ + h^ + r^ = 0,
where (by what precedes) /3 is the vector of the centre, and b the
radius, of a fixed sphere
{p-l3)^ + b^ = 0,
which is cut at right angles by all the varying spheres. By effect-
ing the elimination exactly we easily find b and y3 in terms of given
quantities.
234.J To inscribe in a given sphere a closed polygon, plane or
gauche, whose sides shall be parallel respectively to each of a series of
given vectors.
Let Tp = 1
be the sphere, a, fi, y , -q, 6 the vectors, n in number, and let
Pi,P2, p„ , be the vector-radii drawn to the angles of the polygon.
Then p2~Pi = ^i"' ^^-f ^^■
From this, by operating by S.{p2 + Pi), we get
P2-Pi = = Sap2 + Sapi.
Also = Vap2— Fapi.
Adding, we get = apa + -^"Pi = "Pz + Pi «•
Hence P2=— a~Vi"-
[This might have been written down»at once from the result of
§106.]
Similarly p^ = — /3~V2/3 = ^"^ °-~^ Pi<^^> ^^•
Thus, finally, since the polygon is closed,
P»+i = Pi = i-T&'^rj-^ ^-''a-'p^a^ r,B.
A
128 QUATERNIONS. [235.
We may suppose the tensors of a, )3 t;, 6 to be each unity.
Hence, if ^ ^ „^ ^g^
we have ffl-i = fl-i jj-i /3-1 a-\
which is a known quaternion ; and thus our condition becomes
Pi = (-)"«">]«•
This divides itself into two cases, according as n is an even or an
odd number.
If n be even, we have
api = pya.
Removing the common part p-^^Sa, we have
Fp^Va = 0.
This gives one determinate direction, ± Fa, for ^ ; and shews that
there are two, and only two, solutions.
If n be odd, we have ap^ = —p^a,
which requires that we have
Sa = 0,
i. e. a must be a vector.
Hence Sap^ = 0,
and therefore pj^ may be drawn to any point in the great circle of
the unit-sphere whose poles are on the vector a.
235.] To illustrate these results, let us take first the ease of m= 3.
Here we must have S.aBy =
or the three given vectors must" (as is obvious on other grounds) be
parallel to one plane. Here afiy, which lies in this plane, is (§ 106)
the vector-tangent at the first corner of each of the inscribed tri-
angles; and is obviously perpendicular to the vector drawn from
the centre to that corner.
Ifn=4, we have p^ y f . ajSyb,
as might have been at once seen from §106.
236.] Hamilton has given {Lectures, p. 674) an ingenious and
simple process by which the above investigation is rendered ap-
plicable to the more difficult problem in which each side of the
inscribed polygon is to pass through a given point instead of being
parallel to a given line. His process depends upon the integration
of a linear equation in finite differences. By an immediate appli-
cation of the linear and veetor function of Chapter V, the above
solutions may be at once extended to any central surface of the
second order.
237.] To find the equation of a cone of revolution, whose vertex is
the origin.
240.J THE SPHERE AND CYCLIC CONE. 129
Suppose a, where jTa = l , to be its axis, and e the cosine of its
semi-vertical angle ; then, if p be the vector of any point in the
cone, SaUpz^^e,
or S^ap = —e^p^.
238.] Change the origin to the point in the axis whose vector is
xa, and the equation becomes
{ — X + SaTjrY ^—e^i^a + 'ury.
Let the radius of the section of the cone made by
Saur =
retain a constant value &, while m changes ; this necessitates
X
Vb^ + m^
so that when x is infinite, e is unity. In this case the equation
becomes ^2„^ ^. ^2 ^ j2 _ q^
which must therefore be the equation of a circular cylinder of radius
b, whose axis is the vector a. To verify this we have only to notice
that if w be the vector of a point of such a cylinder we must (§201)
have TFaTu- = b,
which is the same equation as that above.
239.] To find, generally, the equation of a cone which has a circular
section : —
Take the origin as vertex, and let the circular section be the
intersection of the plane Sap = 1
with the sphere (passing through the origin)
p2 = Sl3p.
These equations may be written thus,
SaUp= =-,
-Tp = S^Up.
Hence, eliminating Tp, we find the following equation which Up
must satisfy— SaUpSfiUp =-l,
or p^—SapS^p = 0,
which is therefore the required equation of the cone.
As a and /S are similarly involved, the mere form of this equation
proves the existence of the subcontrary section discovered by Apol-
lonius.
240.] The equation just obtained may be written
S.UaUpS.Ul3Up = --^,
130
QUATERNIONS. [24 1.
or, since a and y3 are perpendicular to the cyclic arcs (§ 59*),
sinj» sinj!)'= constant,
where j) and j)' are arcs drawn from any point of a spherical conic
perpendicular to the cyclic arcs. This is a well-known property of
such curves.
241 .J If we cut the cyclic cone by any plane passing through
the origin, as gyp _ q^
then Fay and Ffiy are the traces on the cyclic planes, so that
p = xUVay+yUF^y (§ 29).
Substitute in the equation of the cone, and we get
—x^—^^ + Pxy = 0,
where P is a known scalar. Hence the values of x and _y are the
same pair of numbers. This is a very elementary proof of the
proposition in § 59*, that PL = MQ (in the last figure of that
section).
243.] When x and ^ are equal, the transversal arc becomes a
tangent to the spherical conic, and is evidently bisected at the
point of contact. Here we have
P=2 = 2S.UrayUrfiy+-^^-''^^'^'
T.VayT^y
This is the equation of the cone whose sides are perpendiculars
(through the origin) to the planes which touch the cyclic cone, and
from this property the same equation may readily be deduced.
243.] It may be well to observe that the property of the Stereo-
graphic projection of the sphere, viz. that the projection of a circle
is a circle, is an immediate' consequence of the above form of the
equation of a cyclic cone.
244 J That § 239 gives the most general form of the equation
of a cone of the second order, when the vertex is taken as origin,
follows from the early results of next Chapter. For it is shewn
in § 249 that the equation of a cone of the second order can always
be put in the form 2 2.Sap8^p + Ap^ = 0.
This may be written 8p<pP = 0,
where <p is the self-conjugate linear and vector function
(^/) = 2F.ap0 + (A + ^Safi)p.
By § 168 this may be transformed to
<i>P=pp+ F. Kpp.,
and the general equation of the cone becomes
{j)-S\p.)p'^ + 2S\pSf^p = 0,
which is the form obtained in § 239.
247-] THE SPHERE AND CYCLIC CONE. 131
•
245.] Taking the form Spct>p =
as the simplest, we fiad by differentiation
Sdp(f>p + Spd<pp = 0,
or '2Sdp(j)p = 0.
Hence (pp is perpendicular to the tangent-plane at the extremity of
p. The equation of this plane is therefore (■nr being the vector of
any point in it) Scj^p (t^-p) = 0,
or, by the equation of the cone,
aSct(^P = 0.
246.] T^e equation of the cone of normals to the tangent-planes of
a given cone can he easily formed from that of the cone itself. For we
may write it in the form
S{<i>-^4,p)<pp = o,
and if we put <pp-=a; a vector of the new cone, the equation becomes
■Sa4>-^<T = 0.
Numerous curious properties of these connected cones, and of the
corresponding spherical conies, follow at once from these equations.
But we must leave them to the reader.
247.] As a final example, let vls find the equation of a cyclic cone
when five of its vector-sides are given — i. e. find the cone of the second
order whose vertex is the origin, and on whose surface lie the vectors
a, A y, S, e.
If we write
= s.r{rapvbi)r(r^yrep)F{rybFpai (i)
we have the equation of a cone whose vertex is the origin — ^for the
equation is not altered by putting sep for p. Also it is the equation
of a cone of the second degree, since p occurs only twice. Moreover
the vectors a, ^,y, 6, e are sides of the cone, because if any one of
them be put for p the equation is satisfied. Thus if we put /3 for p
the equation becomes
= s.v{rafirbe)r{rpyn^)r{rybr^a)
= S.FiFa^ne) { F^aS.FyhF^yFe^- FybS.FfiaFPyFe^}.
The first term vanishes because
S.F{Fa^Fbe)Fl3a= 0,
and the second because
S.F^aF^yFflS = 0,
since the three vectors FjSa, FjSy, Fej3, being each at right angles to
/3, must be in one plane.
As is remarked by Hamilton, this is a very simple proof of Pascal's
K 2,
132 QUATERNIONS.
Theorem — for (1) is the condition that the intersections of the
planes of a, /3 and 8, e ; /3, y and e, p; y, 8 and p, a ; shall lie in one
plane ; or, making the statement for any plane section of the cone,
that the points of intersection of the three pairs of opposite sides, of
a hexagon inscribed in a curve, may always lie in one straight line,
the curve must he a conic section.
EXAMPLES TO CHAPTER VII.
1 . On the vector of a point P in the plane
Sap= 1
a point Q is taken, such that QO.OP is constant ; find the equation
of the locus of Q.
2. "What spheres cut the loci of P and Q in (1) so that both
lines of intersection lie on a cone whose vertex is ?
3. A sphere touches a fixed plane, and cuts a fixed sphere. If
the point of contact with the plane be given, the plane of the inter-
section of the spheres contains a fixed line.
Find the locus of the centre of the variable sphere, if the plane of
its intersection with the fiied sphere passes through a given point.
4. Find the radii of the spheres which touch, simultaneously, the
four, given planes
Sap = 0, Sj3p = 0, Syp = 0, Sbp = 1.
[What is the volume of the tetrahedron enclosed by these planes ?]
5. If a moveable line, passing through the origin, make with
any number of fixed lines angles 6, 6^, 02, &c., such that
a cos.O + «! cos.^i + = constant,
where «, «i, are constant scalars, the line describes a right cone.
6. Determine the conditions that
Sp(j)p ^
may represent a ri^M cone.
7. What property of a cone (or of a spherical conic) is given
directly by the following form of its equation,
S.ipxp ^ ?
8. What are the conditions that the surfaces represented by
Sp^p = 0, and S.ipKp = 0,
may degenerate into pairs of planes ?
EXAMPLES TO CHAPTER VII. 133
9. Find the locus of the vertices of all right cones which have a
common ellipse as base.
10. Two right circular cones have their axes parallel, shew that
the orthogonal projection of their curve of intersection on the plane
containing their axes is a parabola.
11. Two spheres being given in magnitude and position, every
sphere which intersects them in given angles will touch two other
fixed spheres and cut a third at right angles.
12. If a sphere be placed on a tablcj the breadth" of the elliptic
shadow formed by rays diverging from a fixed point is independent
of the position of the sphere.
1 3. Form the equation of the cylinder which has a given circular
section, and a given axis. Find the direction of the normal to the
subcontrary section.
14. Given the base of a spherical triangle, and the product of
the cosines of the sides, the locus of the vertex is a spherical conic,
the poles of whose cyclic arcs are the extremities of the given
base.
15. (Hamilton, Bishop Law's 'Premium Ex., 1858.)
(a.) What property of a sphero-conic is most immediately in-
dicated by the equation
5^5^=1?
a p
{b.) The equation {VKpf + {StipY =
also represents a cone of the second order ; A. is a focal
line, and jj. is perpendicular to the director-plane cor-
responding.
(c.) What property of a sphero-conic does the equation most
immediately indicate ?
16. Shew that the areas of all triangles, bounded by a tangent
to a spherical conic and the cyclic arcs, are equal.
17. Shew that the locus of a point, the sum of whose arcual dis-
tances from two given points on a sphere is constant, is a spherical
conic.
18. If two tangent planes be drawn to a cyclic cone, the four
lines in which they intersect the cyclic planes are sides of a right
cone.
19. Find the equation of the cone whose sides are the intersections
of pairs of mutually perpendicular tangent planes to a given cyclic
cone.
134 QUATERNIONS.
20. Find the condition that five given points may lie on a
sphere.
21. What is the surface denoted by the equation
where p = xa+y^ + zy,
a, )3, y being given vectors, and x, y, z variable scalars ?
Express the equation of the surface in terms of p, a, /3, y alone.
22. Find the equation of the cone whose sides bisect the angles
between a fixed line and any line, in a given, plane, which meets the
fixed line.
What property of a spherical conic is most directly given by
this result ?
CHAPTER VIII.
SURFACES OF THE SECOND OEDEB.
248.] The general scalar equation of the second order in a vector
p must evidently contain a term independent of p, terms of the form
S.apb involving p to the first degree, and others of the form S.aphpc
involving p to the second degree^ a, h, c, &e. being constant quater-
nions. Now the term S.apd may be written as
SpF(da),
or as S.{Sa+ ra)p{Sb+ Vb) = SaSpFb + SbSpFa + S-pFbra,
each of which may evidently be put in the form Syp, where y is a
known vector.
Similarly * the term S.apbpc may be reduced to a set of terms,
each of which has one of the forms
Ap^, [Sapf, SapSpp,
the second being merely a particular case of the third. Thus (the
numerical factors 2 being introduced for convenience) we may write
the general scalar equation of the second degree as follows : —
2S.SapS0p + Ap'^ + 2Syp = a (1)
249.] Change the origin to 1) where OJD = 6, then p becomes
p + b, and the equation takes the form
22.SapS0p + Ap^+21(SapSpb + SfipSab) + 2AS&p+2Syp
+ 2-S.SabSl3b + Ab^ + 2Syb—C=0;
from which the first power of p disappears, that is tie surface is
referred to its centre, if
2(o-S'y38 + ;3<S'a8) + J5 + y = 0, (2)
• For S.aphpc=S.capip=S.a'php = (2Sa'Sb—Sa'b)p' + 2Sa'p8bp; and in particular
cases we may have Va'= Vb.
136 QUATERNIONS. [25O.
a vector equation of the first degree^ wLicli in general gives a single
definite value for 8, by the processes of Chapter V. [It would lead
us beyond the limits of an elementary treatise to consider the
special cases in which (2) represents a line, or a plane, any point of
which is a centre of the surface. The processes to be employed in
such special cases have been amply illustrated in the Chapter re-
ferred to.]
With this value of 6, and putting
the equation becomes
2'L.SapSpp + Ap^=I).
If 2? =^ 0, the surface is conical (a case treated in last Chapter) ;
if not, it is an ellipsoid or hyperboloid. Unless expressly stated not
to be, the surface will, when B is not zero, be considered an ellip-
soid. By this we avoid for the time some rather delicate con-
siderations.
By dividing by B, and thus altering only the tensors of the
constants, we see that the equation of central surfaces of the second
order, referred to the centre, is (excluding cones)
2^{Sap8fip)+gp' = \ (3)
250.] Differentiating, we obtain
2'S{SadpSfip + SapS^dp} + 2gSpdp = 0,
or 8.dp{1{a8pp + pSap) +gp} = 0,
and therefore, by § 137, the tangent plane is
<S(ot-p) {■2{cuS^p + pSap)+gp} = 0,
i.e. S.'!!T{l(aSl3p + pSap)+ffp} = 1, by (3).
Hence,if v = l{aSfip + pSap) + ffp, (4)
the tangent plane is Svur = 1,
and the surface itself is Si>p = 1.
And, as v'^ (being perpendicular to the tangent plane, and satis-
fying its equation) is evidently the vector-perpendicular from the
origin on the tangent plane, v is called the vector ofpronmity.
251.] Hamilton uses for v, which is obviously a linear and vector
function of p, the notation ^p, expressing a functional operation,
as in Chapter V. But, for the sake of clearness, we will go over
part of the ground again, especially for the benefit of students who
have mastered only the more elementary parts of that Chapter.
We have, then, (fip z=2{aSpp+^Sap)+ffp.
2 53-j SURFACES OF THE SECOND ORDEE. 137
With this definition of (f>, it is easy to see that
(«.) (j>{p + a-) = (f)p + <f>(T, &e., for any two or more vectors.
(5.) (f) (a;/)) = :e(l>p, a particular case of (a), x being- a scalar,
(c.) d(f>p = (l>{dp).
{d.) Scr(^p = l,{SacTSfip + S^<TSap)+ffSp(T = Spcpa;
or <p is, in this ease, self-conjugate.
This last property is of great importance.
252.] Thus the general equation of central surfaces of the second
degree (excluding cones) may now be written
Sp4>P=l (1)
Differentiating, Sdpipp + Spd(j>p = 0,
which, by applying (c.) and then (d.) to the last term on the left,
gives 2S^pdp=Q,
and therefore, as in § 250, though now much more simply, the
tangent plane at the extremity of p is
5(^-p)# = 0,
or Stit^p := Sp(f>p = 1.
If this pass through A{OA = a), we have
Saipp = 1,
or, by (d.), Spcfia = 1,
for all possible points of contact.
This is therefore the equation of the plane of contact of tangent
planes drawn from J.
253.] To find the enveloping cone whose vertex is A, notice that
{Sp4>p-l)+j){Sp4>a-lf = 0,
where p is any scalar, is the equation of a surface of the second
order touching the ellipsoid along its intersection with the plane.
If this pass through A we have
{Sa^a—\)-irp{Sa4,a.+ Vf = 0,
and p is found. Then our equation becomes
{Sp^p-l){Sa(j>a-l)—{Sp(j)a—lf = 0, (1)
which is the cone required. To assure ourselves of this, transfer
the origin to A, by putting p + a for p. The result is, using {a.)
and (d.),
{Sp(l)p+2Sp^a + Sa(j}a—l){Sa(j)a—l) — {Sp(pa + Safl}a-lf = 0,
or Sptpp {Sacfia — 1 ) — (Sp(j)aY = 0,
which is homogeneous in Tp^, and is therefore the equation of a
cone.
138 QUATEENIONS. [254.
Suppose A infinitely distant, then we may put in (1) xa for a,
where x is infinitely great, and, omitting all but the higher terms,
the equation of the cylinder formed by tangent lines parallel to a is
{Sp<Pp—l)Sa<i>a—{8p^af = 0.
254.J To study the nature of the surface more closely, let us
find the locus of the middle joints of a system of parallel chords.
Let them be parallel to a, then, if ot be the vector of the middle
point of one of them, ^a + xa and isr — xa are simultaneous values of
p which ought to satisfy (1) of § 252.
That is S.{'!!y±xa)i^{ts±xa)= \.
Hence, by {a.) and {d.), as before.
Surd's + x^Sa<j)a = 1,
S'ST(l>a=zO (1)
The latter equation shews that the locus of the extremity of ot,
the middle point of a chord parallel to a, is a plane through the
centre, whose normal is (pa ; that is, a plane parallel to the tangent
plane at the point where OA cuts the surface. And {d.) shews that
this relation is reciprocal — so that if /3 be any value of w, i. e. be
any vector in the plane (1), a will be a vector in a diametral plane
which bisects all chords parallel to /3. The equations of these
planes are Sw^a = 0,
S-ai^fi = 0,
so that if F. ^a^/3 = y (suppose) is their line of intersection, we have
Sycpa = = Sacj>y, \
Sy<t>^ = = Sfi,i>yA (2)
and (1) gives Sficpa = = Sacp/B. )
Hence there is an infinite number of sets of three vectors a, /3, y,
such that all chords parallel to any one are bisected by the diametral
plane containing the other two.
255.] It is evident from § 23 that any vector may be expressed
as a linear function of any three others not in the same plane, let
then p = xa+yfi + zy,
where, by last section, Sa^/3 = Sficpa = 0,
Satpy = Sycpa = 0,
Sl3(j)y = Sy<l>l3 = 0.
And let Sacpa = 1. )
S/3ct,l3 = 1, [
Sycpy = 1, )
so that a, /3, and y are vector conjugate semi-diameters of the surface
we are engaged on.
2 57-] SURFACES OF THE SECOND ORDEE. 139
Substituting the above value of p in the equation of the surface,
and attending to the equations in a, /3, y and to (a.), {b.), and (cL),
we have Sp<l)p = S{m + i/fi + zy) ^ {osa +yfi + zy),
= x^ +y2 + z^ = 1 .
To transform this equation to Cartesian coordinates, we notice that
X is the ratio which the projection of p on a bears to a itself, &c.
If therefore we take the conjugate diameters as axes of f, j;, f, and
their lengths as a, b, c, the above equation becomes at once
^2 -I- §2 + g2
the ordinary equation of the ellipsoid referred to conjugate diameters.
256.] If we write —^^ instead of ^, these equations assume an
interesting form. We take for granted, what we shall afterwards
prove, that this halving or extracting the root of the vector func-
tion is lawful, and that the new linear and vector function has the
same properties («.), {b.), (c), {d.) (§ 251) as the old. The equation
of the surface now becomes
Sp^l,^p = -l,
or ^^P^P = — 1)
or, finally, T^p = 1.
If we compare this with the equation of the unit-sphere
Tp=l,
we see at once the analogy between the two surfaces. TAe sphere
can be changed into the ellipsoid, or vice versa, by a linear deformation
of each vector, the operator being the function yjr or its inverse. See
the Chapter on Kinematics.
257.] Equations (2) § 254 now become
Sa\l/^I3= =S\j,a\j/^, &c., (1)
so that yj/a, \lf^, \(ry, the vectors of the unit-sjahere which correspond to
semi-conjugate diameters of the ellipsoid, form a rectangular system.
We may remark here, that, as the equation of the ellipsoid referred
to its principal axes is a case of § 255, we may now suppose i,j, and
3tj TJ Hy
k to have these directions, and the equation is -^ + j^ -^ — 2 = ^j
which, in quaternions, is
{SipY {Sjpf {Skpf _
Sp<i>P=-^ + -^ + —^- - 1-
We here tacitly assume the existence of such axes, but in all cases,
by the help of Hamilton's method, developed in Chapter V, we at
once arrive at the cubic equation which gives them.
140 QUATERNIONS. [258.
It is evident from the last-written equation that
iSip jSjp kSkp
and
a'' b'' c"
^ V a b '
which latter may be easily proved by shewing that
And this expression enables us to verify the assertion of last section
about the properties of ■^.
As 8ip=. —X, &c., x,y, z being the Cartesian coordinates referred
to the principal axes, we have now the means of at once transform-
ing any quaternion result connected with the ellipsoid into the or-
dinary one.
258.] Before proceeding to other forms of the equation of the
ellipsoid, we may use those already given in solving a few problems.
Find ike locus of a point when the perpendicular from the centre on
its polar plane is of constant length.
If OT be the vector of the point, the polar plane is
Spt^T^ = 1,
and the length of the perpendicular from is ^f- — (§ 208).
Hence the required locus is
T4>^ = G,
or ^OT()!)V=-C2,
a concentric ellipsoid^ with its axes in the same direction as those
of the first. By § 257 its Cartesian equation is
259.] Find the locus of a point whose distance from a given point
is always in a given ratio to its distance from a given line.
Let p=xj3 be the given line, and A{OA=a) the given point, and
let Safi = 0. Then for any one of the required points
Tip-a) = eTrpp,
a surface of the second order, which may be written
p^-2Sap+a^ = e2 (6'2/3p_/3V)-
Let the centre be at 8, and make it the origin, then
p^ + 2Sp{b-a) + {b-af = e^S^.^{p + b)-fi^{p-\-by},
and, that the first power of p may disappear,
{b-a) = e^{l3Sl3b-l3^),
a linear equation for 6. To solve it, note that <Sa/3 = 0, operate by
S.^ and we get (1 -e^/S^ + e^^^)S^b = S^b = 0.
2 6 1. J SURFACES OF THE SECOND ORDER. 141
Hence 8-a = -e^\
or
Referred to this point as origin the equation becomes
which shews that it belongs to a surface of revolution (of the second
order) whose axis is parallel to /3, as its intersection with a plane
S^p = a, perpendicular to that axis, lies also on the sphere
P'
e^a^ e^/3^a^
H-e2/32 {1 + e^^y
In fact, if the point be the focus of any meridian section of an
oblate spheroid, the line is the directrix of the same.
260.] A sphere, jiassing through the centre of an ellipsoid, is cut hy
a series of spheres whose centres are on the ellipsoid and which pass
through the centre thereof; find the envelop of the planes of inter-
section.
Let [p — df = o^ be the first sphere, i.e.
p^ — 2Sap= 0.
One of the others is p^ — 2&3-p = 0,
where Snrcjyss- = 1 .
The plane of intersection is
S{7s — a)p = 0.
Hence, for the envelop, (see next Chapter,)
S'sr d>nr = 0, ) , , ,
„ , „ > where cr = afar,
S'urp = 0, )
or <^OT = xp, {Vx = 0},
i.e. CT = co(l)~'^p.
Hence x^Sp^-^.p =1, 1
and xSp<l)~'^p = Sap, )
and, eliminating x,
Sp,j>-^p = {Sap)^
a cone of the second order.
261. J From a point in the outer of two concentric ellipsoids a tan*
gent cone is d/rawn to the inner, find the envelop of the plane of contact.
If Si!r(f>zT = 1 be the outer, and iSp^p = 1 be the inner, <f) and -^
being any two self-conjugate linear and vector functions, the plane
of contact is Surxj/p = 1. .
Hence, for the envelop, Sm'^p = 0,
Sm\
tt'^P = 0, )
3-'(^CT =: 0, )
142 QUATERNIONS. [262.
therefore (^ot = a!\//p,
or tn- = x<^~^-\\ip.
This gives xS.^p(^~'^^p = !> )
and x'^S.^p(\>~^-^p = 1, )
and therefore, eliminating x,
S.^lrp,f>-^^jrp=-i,
or S.p\j/tj)~^-^p = 1,
another concentric ellipsoid, as \jf(l)~^\jf is a linear and vector func-
tion = \ suppose ; so that the equation may be written
Spxp= 1.
263.] Find the locus of intersection of tangent planes at the extre-
mities of conjugate diameters.
If a, /3, y be the vector semi-diameters, the planes are
8vr]f'^a= — \, •\
with the conditions § 257.
Hence —^^v!S.-^w\i^^y=^'ss = ^a-\--<^^-V'^y, by § 92,
therefore T^ts = vS,
since yjra, ^jS, \jfy form a rectangular system of unit- vectors.
This may also evidently be written
fci/^^^ = - 3,
shewing that the locus is similar and similarly situated to the given
ellipsoid, but larger in the ratio -s/s : 1 .
263.] ' Find the locus of the intersection of three sjiheres whose dia-
meters are semi-conjugate diameters of an ellipsoid.
If a be one of the semi-conjugate diameters
Sa\l/^a = — 1.
And the corresponding sphere is
p^—Sap=0,
or p^—S\^ai^~^p = 0,
with similar equations in /3 and y. Hence, by § 92,
y}f-^pS.\jra\j/^\j/y = -i'-'^p = p'^{\lfa + \l/l3 + \l/y),
and, taking tensors, T^'^p = VsTp^,
or ^-^"^=^3,
or, finally, Sprj/'^p ;^-3p\
This is Fresnel's Surface of Elasticity in the Undulatory Theory.
264.] Before going farther we may prove some useful properties
of the function ^ in the form we are at present using — viz.
iSip jSjp kSkp
265.] SURFACES OP THE SECOND OKDER.
143
We have p =
and it is evident that
(jii =■
^J = -i
# = -^2'
Hence
_ iSip jSjp kSkp
<^V=-^-
a* b* C
<j>~^P = aHSip + bySjp + c^/cSkp,
Also
and so on.
Again, if a, /3, y be any rectangular unit-vectors
But as
we have
Again,
„2 ^ ^2
&c. = &c.
(Sipf + {Sjp)^ + {Skpf=-p\
Sa(f,a + Sfi<l>^ + Sy<py = 1^ + ^+1^
S.
ASia
iSi^
ASiy
*.♦,*,=«.(= + ...)(5^ + ...)('2?+...)
Sia
Siy
,2
b^ '■
b^ ■
Sjy
Ska
c2
Sk^
c2
Sky
— 1
a^b^c'
Sia, Sja, Ska
Sip, Sjl3, Skp
Siy, Sjy, Sky
= +
a^b^c^
And so on. These elementary investigations are given here for the
benefit of those who have not read Chapter V. The student may
easily obtain all such results in a far more simple manner by means
of the formulae of that Chapter.
265.] MnAthe locus of intersection of a rectangular system of three
tangents to an ellipsoid.
If tn- be the vector of the point of intersection, a, /3, y the tangents,
then, since •m + xa should give equal values of a; when substituted in
the equation of the surface, giving
S {m + Xa) <p {-or + xa) = 1,
or x^Sa(\)a + 2xS^(f>a + (/Soti^ct — 1 ) = 0,
we have {S^ipaY = Sa<l)a {S-sr(j)w—l).
Adding this to the two similar equations in /3 and y
(/Sa^ti7)2 + (S/Scp^f + {Sy(l)wf = {Sa^a + <S/3<^/3 + Sy<l>y) (/Stsrc/.w - 1 ),
144 QUATERNIONS. [266.
or -{<}>^f = (1 + 1, + ^) {S:^^-l),
an ellipsoid concentric with the first.
366.] If a rectangular system of chords he drawn through any point
within an ellipsoid, the sum of the reciprocals of the rectangles under
the segments into which they are divided is constant.
With the notation of the solution of the preceding problem, w
giving the intersection of the vectors, it is evident that the product
of the values of x is one of the rectangles in question taken nega-
tively.
Hence the required sum is
1 £ 1
Sta^TH — 1 Ssy^'ST — 1
This evidently depends on Smcfrar only and not on the particular
directions of a, ^,y : and is therefore unaltered if ■nr be the vector
of any point of an ellipsoid similar, and similarly situated, to the
given one. [The expression is interpretable even if the point be
exterior to the ellipsoid.]
267.] Shew that if any rectangular system of three vectors he drawn
from a point of am, ellipsoid, the plane containing their other extremities
passes through a jimed point. Find the locus of the latter point as tlie
former varies.
With the same notation as before, we have
SsT(j)Zl7 ^ 1,
and 8 (;sr + X a) (j) (tn- + xa) = 1 ;
, , „ 2Sa<b-sr
thereiore x = •
ocupa
Hence the required plane passes through the extremity of
Sa(pa
and those of two other vectors similarly determined. It therefore
passes through the point whose vector is
aSa^Tjy + ^SlB^yar + ySycjiZT
Sa(t>a + S^(l)l3 + 8y(l>y '
or 6 = ^+^-^ (§173).
Thus the first part of the proposition is proved.
268.] SURFACES OF THE SECOND ORDER. 145
OTo
-1
But we have also ot = — ("(^ + — )
whence by the equation of the ellipsoid we obtain
the equation of a concentric ellipsoid.
268.] Find the directions of the three vectors which are parallel to
a set of conjugate diameters iti each qf two central surfaces of the second
degree.
Transferring the centres of both to the origin, let their equations
be Sp(t>p.— 1 or 0,;
and Sp\l/p= 1 or O.S ^'
If a, l3, y be vectors in the required directions, we must have (§254)
Sa(p^ = 0, Sa\{/^ = 0, \
S^<t>y^Q, S^^lry=Q^ (2)
Sy(l>a = 0, Syfa = 0. )
From these equations 0a || V^y || ^a, &c.
Hence the three required directions are the roots of
r.<t>pi'p = o (3)
This is evident on other grounds, for it means that if one of the
surfaces expand or contract uniformly till it meets the other, it will
touch it successively at points on the three sought vectors.
We may put (3) in either of the following forms —
F.p4,-^irp=0,]^
or r.p\/f-i(|)p= 0;i ^ ''
and, as <j) and v/f are given functions, we find the solutions by the
processes of Chapter V.
[iVbfe. As (j)~^^ and V~^^ ^^^ ^°^> ^^ general, self-conjugate
functions, equations (4) do not signify that a, /3, y are vectors parallel
to the principal axes of the surfaces
<S.p0-Vp = 1> S.p^jf-^(t)p = 1.
In these equations it does not matter whether (j)~^^ is self-conjugate
or not ; but it does most particularly matter when they are differ-
entiated, so as to find axes, &c.]
Given two surfaces of the second degree, there exists in general a set
of Cartesian axes, whose directions are those of conjugate diameters in
every one of the surfaces of the second degree passing through the inter-
section of the two surfaces given.
L
146 QUATERNIONS. [269.
For any surface through the intersection of
Sp(j)p=l and S{p—a)^{p—a) = e,
is fSp4>p—8{p—a)-f{p — a)=f—e,
where/ and e are scalars.
The axes of this depend only on the term
Sp{fct>-y},)p.
Hence the set of conjugate diameters which are the same in all are
the roots of
J'i/'t>-^)pU'i4>-^)p=0, or rcpp^p=0,
as we might have seen without analysis.
The locus of the centres is given by the equation
('/'-/^)P-V'« = o>
where/" is a scalar variable.
269.] Find the equation of the ellipsoid of which three conjugate
semi-diameters are given.
Let the vector semi-diameters be a, j3, 7, and let
8p4,p = 1
be the equation of the ellipsoid. Then (§ 255) we have
Sa(i)a = \, Sa(pfi = 0,
Sycjyy = 1, Sy^a ^ ;
the six scalar conditions requisite (§ 139) for the determination of
the linear and vector function (j).
They give a \\ V(j)^(j)y,
or xa = (j}~^ F/3y.
Hence ■ cc = ccScupa = S.afiy,
and similarly for the other combinations. Thus, as we have
pS.a^y = cuS.^yp+^S.yap + yS.afip,
we find at once
<j)pS^.al3y = Fl3yS.j3yp + VyaB.yap+ FafiS.a^p;
and the required equation may be put in the form
S^.afiy = S^.a^p + S^.fiyp + S^.yap.
The immediate interpretation is that if four tetrahedra be formed iy
growping, three and three, a set of semi-conjugate vector axes of an
ellipsoid and any other vector of the surface, the sum of the squares of
the volumes of three of these tetrahedra is equal to the square of the
volume of the fourth.
2 7 2. J SURFACES OP THE SECOND ORDER. 147
•
370.] When the equation of a surface of the second order can be
put in the form Sp(()-''-p = I, (1)
where (<^-^)(<#>-^i)(<l'-^2) = 0. _
we know that ff, ff^ , g^ are the squares of the principal semi-diameters.
Hence, if we put (|) + ^ for <^ we have a second surface, the diifer-
enees of the squares of whose principal semiaxes are the same as for
thefirst. Thatis, 8p{<^ + h)-'^p=\ (2)
is a surface confocal with (1). From this simple modification of the
equation all the properties of a series of confocal surfaces may easily
be deduced. We give one as an example.
271.] Any two confocal surfaces of the second order, wJiich meet,
intersect at right angles.
For the normal to (2) is, evidently,
and that to another of the series, if it passes through the common
point whose vector is p, is there
(<^ + /ii)-V.
But ^.(<^+^)-v(^+^o-P = ^•P (^^,)(^^^y
and this evidently vanishes if h, and h-^ are different, as they must be
unless the surfaces are identical.
272.] To find the conditions of similarity of two central surfaces
of the second order.
Referring them to their centres, let their equations be
8p<^'p=\.\ ^'^
Now the obvious conditions are that the axes of the one are pro-
portional to those of the other. Hence, if
g^-\-m^g'^ + m^g ^m=fi,\
^g'nm\g'+m'=0,i ^'^
/' +
be the equations for determining the squares of the reciprocals of
the semiaxes, we must have
—^=IJ; -^ = IJ.^, — = IJ,^, (3)
m^ m^ m '
where \x. is an undetermined scalar. Thus it appears that there are
but two scalar conditions necessary. Eliminating jn we have
ni'\ _ nn'y m'm\ _ m'\
m% ~ %' mm^ ~ mf ^ ■'
which are equivalent to the ordinary conditions.
L a
148 QUATERNIONS. [273.
273.] Find. the greatest and least semi-diameters of a central plane
sectioti of an ellipsoid.
Here Spcl,p = I I
Sap=o] ^ >
together represent the elliptic section ; and our additional condition
is that Tp is a maximum or minimum.
Differentiating the equations of the ellipse, we have
S(f>pdp = 0,
Sadp = 0,
and the maximum condition gives
dTp = 0,
or Spdp = 0.
Eliminating the indeterminate vector dp we have
S.apcf>p = (2)
This shews that tAe maximum or minimum' vector, the normal at its
extremity, and the perpendicular to the plane of section, lie in one
plane. It also shews that there are but .two vector-directions which
satisfy the conditions, and that they are perpendicular to each other,
for (2) is satisfied if ap be substituted for p.
We have now to solve the three equations (1) and (2), to find the
vectors of the two (four) points in which the ellipse (1) intersects
the cone (2). We obtain at once
4>p = xV.<^~'^dVap.
Operating by S.p we have
1 = xp^Sa(l)~^a.
XT 2 J. Sp(j)-''-a
Hence p'op = p-a „ , ^
»' '=^('-''«--> «
fromwhich ■ S.a{l—p^(f))-^a= ; (4)
a quadratic equation in p^, from which the lengths of the maximum
and minimum vectors are to be determined. By § 147 it may be
written mp*Sa(l)-'^a—p^S.a{m2—(t>)a+a' = (5)
[If we had operated' by 8.<p-^a or by 8.(pr^p, instead of by S.p,
we should have obtained an equation apparently different from this,
but easily reducible to it. To prove their identity is a good exercise
for the student.]
Substituting the values of p^ given by (5) in (3) we obtain the
vectors of the required diameters. [The student may easily prove
directly that {\—pl<f>)-'^a and {l—pl^)-^a
276.] SURFACES OF THE SECOND ORDER. 149
are necessarily perpendicular to each other, if both be perpendicular
to a, and if pf and p| be different. See § 271.]
274.] By (5) of last section we see that
2 2 _ "^
Hence the area of the ellipse (1) is
V — mSa<f)~^a
Also the locus of normals to all diametral sections of an ellipsoid,
whose areas are equal, is the cone
Sa(t>-'^a = Co?.
When the roots of (5) are equal, i.e. when
{m..fl^—Sa^af = ima'^Satp-'^a, (6)
the section is a circle. It is not difficult to prove that this equation
is satisfied by only two "Values of Ua, but another quaternion form
of the equation gives the solution of this and similar problems by
inspection. (See § 275 below.)
275.] By § 168 we may write the equation
Sp<f>p =: 1
in the new form S.Kpfxp + pp^ = 1,
where ^ is a known scalar, and A. and f/. are definitely known (with
the exception of their tensors, whose product alone is given) in
terms of the constants involved in </>. [The reader is referred again
also to §§ 121, 122.] This may be written
2SkpSij.p + {p—SKiJ.)p^ = l (1)
From this form it is obvious that the surface is cut by any plane
perpendicular to A. or fi in a circle. For, if we put
S\p = a,
we have 2aSixp + {p—S\ix)p^ = 1,
the equation of a sphere which passes through the plane curve of
intersection.
Hence X and n o( § 168 are the values of a in equation (6) of the
preceding section.
276.] Any two circular sections of a central surface of the second
order, whose planes are not parallel, lie on a sphere.
For the equation {S\p—a) (Sixp — b) = 0,
where a and b are any scalai* constants whatever, is that of a
system of two non-parallel planes, cutting the surface in circles.
Eliminating the product SKpS^p between this and equation (1) of
last section, there remains the equation of a sphere.
150 QUATEENIONS. [277.
277.] To find the generating lines of a central surface of the second
order.
Let the equation be Spcpp = 1 ;
then, if a be the vector of any point on the surface, and ■nr a vector
parallel to a generating line, we must have
p = a + xm
for all values of the scalar x.
Hence 8 {a + xw) <^ (o + xm) = 1 ,
which gives the two equations
■=o.J
The first is the equation of a plane through the origin parallel to
the tangent plane at the extremity of a, the second is the equation
of the asymptotic cone. The generating lines are therefore parallel
to the intersections of these two surfaces, as is well known.
Froni these equations w.e have
ycfysT = Fota-
where _^ is a scalar to be determined. Operating on this by S.^ and
S.y, where y3 and y are any two vectors not coplanar with a, we have
S^{ycl>^+ra^) = 0, Sm{i/<t,y—rya) = (1)
Hence S.<})a (j^^^ + Fa/3) {y(j)y— Vya) = 0,
or my^S.a^y—SacpaS.a^y = 0.
Thus we have the two values
Sa<f>'sr = 0,
a /I
belonging to the two generating lines.
278.] But by equation (1) we have
zm = r.(y^/3+ Va^) {y^y— Vyd)
= my"^ (j)-^ V^y + yV.^a V^y — aS.aVfiy ;
which, according to the sign of y, gives one or other generating
line.
Here V^y may be any vector whatever, provided it is not per-
pendicular to a (a condition assumed in last section), and we may
write for it 6.
Substituting the value of y before found, we have
zvT = (t)-^d—ajSa0 + ^ — Fd>a0,
278.J SURFACES OF THE SECOND ORDER. 151
or, as we may evidently write it,
= <i>-'^{r.ar4>ae)±J~r^ae (2)
Put r = V^a6,
and we have zur = d>-^ Far + ^— t,
~ ^ m
with the condition Srcpa = 0.
[Any one of these sets of values forms the complete solution of the
problem ; but more than one have been given, on account of their
singular nature and the many properties of surfaces of the second
order which immediately follow from them. It will be excellent
practice for the student to shew that
is an invariant. This may most easily be done by proving that
V.y^e-^Oi = identically.]
Perhaps, however, it is simpler to write a for F/3y, and we thus
«CT- = — d) '■ya yaAa + x/ — Va<i>a.
^ m
[The reader need hardly be reminded that we are dealing with the
general equation of the central surfaces of the second order — the
centre being origin.]
EXAMPLES TO CHAPTER VIII.
1 . Find the locus of points on the surface
Sp<^p = 1
where the generating lines are at right angles to one another.
2. Find the equation of the surface described by a straight line
which revolves about an axis, which it does not meet, but with
which it is rigidly connected.
3. Find the conditions that
Sp^p = 1
may be a surface of revolution, with axis parallel to a given vector.
4. Find the equations of the right cylinders which circumscribe
a given ellipsoid.
5. Find the equation of the locus of the extremities of perpen-
diculars to central plane sections of an ellipsoid, erected at the
152 QUATERNIONS.
centre, their lengths being the principal semi-axes of the sections.
[Fresnel's Wave-Surface. See Chap. XI.]
6. The cone touching central plane sections of an ellipsoid, which
are of equal area, is asymptotic to a confocal hyperboloid.
7. Find the envelop of all non-central plane sections of an ellip-
soid whose area is constant.
8. Find the locus of the intersection of three planes, perpendicular
to each other, and touching, respectively, each of three confocal
surfaces of the second order.
9. Find the locus of the foot of the perpendicular from the centre
of an ellipsoid upon the plane passing through the extremities of a
set of conjugate diameters.
10. Find the points in an ellipsoid where the inclination of the
normal to the radius-vector is greatest.
1 1 . If four similar and similarly situated surfaces of the second
order intersect, the planes of intersection of each pair pass through
a common point.
12. If a parallelepiped be inscribed in a central surface of the
second degree its edges are parallel to a system of conjugate dia-
meters.
13. Shew that there is an infinite number of sets of axes for which
the Cartesian equation of an ellipsoid becomes -
x^-^y'^+z^ = e^.
14. Find the equation of the surface of the second order which
circumscribes a given tetrahedron so that the tangent plane at each
angular point is parallel to the opposite face; and shew that its
centre is the mean point of the tetrahedron.
15. Two similar and similarly situated surfaces of the second
order intersect in a plane curve, whose plane is conjugate to the
vector joining their centres.
16. Find the locus of all points on
Sp(i>p = 1,
where the normals meet the normal at a given point.
Also the locus of points on the surface, the normals at which
meet a given line in space.
17. Normals drawn at points situated on a generating line are
parallel to a fixed plane.
18. Find the envelop of the planes of contact of tangent planes
drawn to an ellipsoid from points of a concentric sphere. Find the
locus of the point from which the tangent planes are drawn if the
envelop of the planes of contact is a sphere.
EXAMPLES TO CHAPTER VIII. 153
19. The sum of the reciprocals of the squares of the perpendiculars
froiQ the centre upon three conjugate tangent planes is constant.
20. Cones are drawn, touching an ellipsoid, from any two points
of a similar, similarly situated, and concentric ellipsoid. Shew that
they intersect in two plane curves.
Find the locus of the vertices of the cones when these plane sec-
tions are at right angles to one another.
2 1 . Find the locus of the points of contact of tangent planes
which are equidistant from the centre of a surface of the second
order.
22. From a fixed point A, on the surface of a given sphere, draw
any chord AB; let 1/ be the second point of intersection of the
sphere with the secant £D drawn from any point £ ; and take a
radius vector AE, equal in length to SB', and in direction either
coincident with, or opposite to, the chord AD : the locus of S is an
ellipsoid, whose centre is A, and which passes through B. (Hamilton,
Elements, p. 227.)
23. Shew that the equation
p (e2_ 1) (e + Saa) = (Sapf - 2eSapSa'p + (Sa'pf + (1 -e^) p\
where e is a variable (scalar) parameter, and a, a' unit- vectors, repre-
sents a system of eonfocal surfaces. {Ibid. p. 644.)
24. Shew that the locus of the diameters of
Sp<pp = 1
which are parallel to the chords bisected by the tangent planes to
the cone Spfp =
is the cone S.p(f>yjf~''-'(f)p = 0.
25. Find the equation of a cone, whose vertex is one summit of
a given tetrahedron, and which passes through the circle circum-
scribing the opposite side.
26. Shew that the locus of points on the surface
Sp<f)p = 1,
the normals at which meet that drawn at the point p=t!r, is on the
cone «S'.(/)— ot) (t)w(j)p = 0,
27. Find the equation of the locus of a point the square of whose
distance from a given line is proportional to its distance from a
given plane.
28. Shew that the locus of the pole of the plane
Sap = 1,
with respect to the surface
Sp(pp = 1,
lf)4 QUATERNIONS.
is a sphere^ if a be subject to the condition
Sacl>-^a = C.
29. Shew that the equation of the surface generated by hnes
drawn through the origin parallel to the normals to
Sp^-'^p = 1
along its lines of intersection with
Sp{<l> + F]--^P=zi,
is m^ —^Sm {4, + ky^-m: = 0.
30. Common tangent planes are drawn to
2S\pSiJ,p + {p—Skij.)p^ = l, and Tp = k,
find the value of A that the lines of contact with the former surface
may be plane curves. What are they, in this case, on the sphere?
Discuss the case of jo^—S^Xix = 0.
31. If tangent cones be drawn to
Sp(j>^P = 1,
from every point of 'Sp'pp = !>
the envelop of their planes of contact is
Sp^^p = 1.
32. Tangent cones are drawn from every point of
S{p — a)(j>{p — a.) = n^,
to the similar and similarly situated surface
Sp4,p = 1,
shew that their planes of contact envelop the surface
{Sa(l)p-lY = n'^Sp(l)p.
33. Pind the envelop of planes which touch the parabolas
p = ai^ + pt, p = aT^ + yr,
where a, (3, y form a rectangular system, and t and t are scalars.
34. Find the equation of the surface on which lie the lines of
contact of tangent cones drawn from a fixed point to a series of
similar, similarly situatedj and concentric ellipsoids.
35. Discuss the surfaces whose equations are
SapS^p = Syp,
and S^ap + S.a^p— \.
36. Shew that the locus of the vertices of the right cones which
touch an ellipsoid is a hyperbola.
37. If oj, Og, ag be vector conjugate diameters of
Sp(f>p = 1,
where ^^— %<^^ +%(^— m = 0,
shew that 2(a^)=--^> 2(Fa, 0,)^= -} S^.(ua,a~= >
^ ' m V 1 Z^ ^ "12 3 ^
and 2 (<ii>af — m^ .
CHAPTER IX.
GEOMETRY OF CUEVES AND SURFACES.
279.] We have already seen (§31 (l)) that the equations
p = ct>t = S.a/it),
and p = (p{i, u) = 1.af{t, u),
where a represents one of a set of given vectors, and /"a scalar func-
tion of scalars t and u, represent respectively a curve and a surface.
We commence the present too brief Chapter with a few of the im-
mediate deductions from these forms of expression. We shall then
give a number of examples, with little attempt at systematic devel-
opment or even arrangement.
280.] What may be denoted by t and u in these equations is, of
course, quite immaterial : but in the case of curves, considered
geometrically, t is most conveniently taken as the length, s, of the
curve, measured from some fixed point. In the Kinematical in-
vestigations of the next Chapter t may, with great convenience, be
employed to denote time.
281.] Thus we may write the equation of any curve in space as
P = <i>^,
where <^ is a vector function of the length, s, of the curve. Of
course it is only a linem- function when, the equation (as in § 31 {I))
represents a straight line.
283.] We have also seen (§§ 38, 39) that
^P ^ A. A.'
is a vector of unit length in the direction of the tangent at the ex-
tremity of p.
At the proximate point, denoted by "s + hs, this unit tangent vector
becomes ^'s + (|)"s 6« + &e.
156 QUATERNIONS. [283.
But., because T<})'s = 1,
we have S.<j)'s (j/'s = 0.
Hence ij/'s is a vector in the osculating plane of the curve, and per-
pendicular to the tangent.
Also, if bd be the angle bet.ween the successive tangents (j/s and
<f/s + (p"s bs + , we have
<^ = ^*'"
so that t&e tensor of <^"s is the reciprocal of the radius of absolute
curvature at the point s.
283.] Thus, if OP = (/>« be the vector of any point P of the
curve, and if C be the centre of curvature at P, we have
and thus OC = <hs jj-
cf) s
is the equation of the locus of the centre of curvature.
Hence also F.cjj'skj/'s or <f>s^"s
is the vector perpendicular to the osculating plane ; and
T^^{4>'sU<^"s)
is the tortuosity of the given curve, or the rate of rotation of its
osculating plane per unit of length.
284.J As an example of the use of these expressions let us fin^
the curve whose curvature and tortuosity are both constant.
We have curvature = T^"s = Tp"= c.
Hence (j)'s(j/'s = p'p"= ca,
where a is a unit vector perpendicular to the osculating plane. This
o r fff . ffo ^ Pi Oct ^j ff ff
pp +p ^=c^— = cc^Up =Cip ,
if Cj represent the tortuosity.
Integrating we get p'p"- g^p'^^^ (1)
where /3 is a constant vector. Squaring both sides of this equation,
we get c2 = cf -/32 - 2 c^Sfip'
(for by operating with S.p' upon (1) we get +c^ = Sj3p'),
or Tj3 = ^/c^+cl
285.] GEOMETRY OF CURVES AND SURFACES. 157
Multiply (1) by p, remembering that
Tp'= 1,
and we obtain _ p" = _ q 4- p'^^
or, by integration, p = c-^s—pP-\-a, (2)
where a is a constant quaternion. Eliminating p', we have
of which the vector part is
p"— p/32 = —cjsfi— Fafi.
The complete integral of this equation is evidently
P = ieos.sT^ + r,sin.sTl3-~{c^sl3+ Faj3), (3)
f and T] being any two constant vectors. We have also by (2),
Sfip = CjS + Sa,
which requires that Sfi^ = 0, Sfirj = 0.
The farther test, that Tp'=l, gives us
-1 = Tl3\i^sin\sT^ + r,^cos\sT^-2Sir,sm.sTl3eos.sTl3)- -/^ •
This requires^ of course^
so that (3) becomes the general equation of a helix traced on a right
cylinder. (Compare § 31 (m).)
285.] The vector perpendicular from the origin on the tangent
to the curve p = rf)«
is, of course, -,Vp'p, or p'Fpp'
(since p' is a unit vector).
To find a common property of curves whose tangents are all equi-
distant from the origin.
Here TFpp'^z c,
which may be written —p^—S^pp'=c^ (1)
This equatiod shews that, as is otherwise evident, every curve on
a sphere whose centre is the origin satisfies the condition. For ob-
viously —p^ = c^ gives Spp'= 0,
and these satisfy (1).
If Spp' does not vanish, the integral of (1) is
VTp^-c^ = s, (2)
an arbitrary constant not being necessary, as we may measure s
from any point of the curve. The equation of an involute which
commences at this assumed point is
-ST = p — sp'.
158 QUATERNIONS. [286.
This gives T^^ = Tp^ + s^ + 2 sSpp'
= Tp^^s''-2s^/Tp^-c^, by(l),
= o\ by (2).
This includes all curves whose involutes lie on a sphere about the origin.
286.] Find the locus of the foot of the ^perpendicular drawn to a
tangent to a right helix from a point in the axis.
The equation of the helix is
p = acos- +/3sin- 4-y*,
a a '
where the vectors a, ^, y are at right angles to each other, and
Ta = Tl3=z h, while aTy = ^a^-h^.
The equation of the required locus isj by last section,
■ar = p'Vpp'
, s a^—l^ . *\ ^/ . s a'^—W' S\ b^
= a (cos — I 5 — ssin-) + fl(sm 5 — «cos-) + y-^-*.
^ a a^ a^ ^ a a^ a' ' a^
This curve lies on the hyperboloid whose equation is
B'^aTn-^-S^^vs-a^S^yw = «*,
as the reader may easily prove for himself.
287.] To find the least distance between consecutive tangents to a
tortuous curve.
Let one tangent be ct = p + xp' ,
then a consecutive one, at a distance hs along the curve, is
^ = p + p'6« + p"g +&c.+y(p' + /'85 + p"'g +...).
The magnitude of the least distance between these lines is, by
§§203,210,
^.(p'8* + p"g+p'"j^+...)C^r.p'(p' + p"6* + p"'g + ...)
~ Trp'p"is
if we neglect terms of higher orders.
It may be written, since p'p" is a vector, and Tp' = 1 ,
^^.C/p'TpV".
But (§133(2)) ^^^ = r^5s=p,p'S.p'py'
Hence pj-,8.Up"rp'p"'
289.J GEOMETRY OP CURVES AND SURFACES. 159
is the small angle, 6</), betwee:rtlie two successive positions of the
osculating plane. [See also § 283.]
Thus the shortest distance between two consecutive tangents is
expressed by the formula bcfibs^
12/ '
vhere r, = -=y-, , is the radius of absolute curvature of the tortuous
curve.
288.] Let us recur for a moment to the equation of the parabola
(§31(/.)) ^ /3<2
P = "'^ + 2 ■
Here p'= {a + fit)-j-,
whence, if we assume Safi = 0,
from which the length of the are of the curve can be derived in
terms of t by integration.
Again, p"=(a+,0£+K|)^-
dH _ d 1 _ dt S.^{a + pi)
^ ds^ ~ ds ' T{a +/3i!) ~ "•" S T{a + ^tf '
and therefore, for the vector of the centre of curvature we have
(§ 283), ^^^f_^§^ -{a^ + ^H^f{-^o? + afiH)-\
which is the quaternion equation of the evolute.
289.] One of the simplest forms of the equation of a tortuous
curve is fl^2 yp
P = -i + '^ + \'
where a, /3, y are any three non-coplanar vectors, and the numerical
factors are introduced for convenience. This curve lies on a para-
bolic cylinder whose generating lines are parallel to y ; and also on
cylinders whose bases are a cubical and a semi-cubical parabola,
their generating lines being parallel to ^ and a respectively. We
have by the equation of the curve
160 QUATERNIONS. 29O.
from which, by 2'/=!, the length of the curve can be fouud In
terms of t ; and
from which p" can be expressed in terms of s. The investigation
of various properties of this curve is very easy, and will be of great
use to the student.
{Note. — It is to be observed that in this equation t cannot stand
for *, the length of the curve. It is a good exercise for the student
to shew that such an equation as
or even the simpler form
p- a^ + ^s^,
involves an absurdity.]
290.] The equation p = <f)^€,
where cf) is a. given self-conjugate linear and vector function, t a
scalar variable, and e an arbitrary vector constant, belongs to a
curious class of curves.
We have at once — = ^' log (jbe,
where \og<p is another self-conjugate linear and vector function,
which we may denote by x- These functions are obviously commu-
tative, as they have the same principal set of rectangular vectors,
hence we may write gp
which of course gives -j^ = x^Pt &c.,
since x does not involve t.
As a verification, we should have
./,-"e = p+^a^ + ^— + &C.
= (i + s^x+|Jx^+ )p
where e is the base of Napier's Logarithms.
This is obviously true if ^" = e*''',
or (jb = gXj
or log = X.
which is our assumption.
[The above process is, at first sight, rather startling, but the
293- GEOMETEY OF CURVES AND SUEFACES. 161
student may easily verify it by writing, in accordance with the
, ilts of Chapter V,
whence ^«e z= —g[aSaf—gl^^S^e—glySye.
He will find at once
X« = —logg^^aSat - hgg^pSfie-loggsySye,
and the results just given follow immediately.]
291.] That the equation
p = (^ (i5, w) = 2 . af{t, u)
represents a surface is obvious from the fact that it becomes the
equation of a definite curve whenever either t ov u has a particular
value assigned to it. Hence the equation at once furnishes us with
two systems of curves, lying wholly on the surface, and such that
one of each system can, in general, be drawn through any assigned
point on the surface. Tangents drawn to these curves at their
point of intersection must^ of course, lie in the tangent plane, whose
equation we have thus the means of forming.
292.] By the equation we have
* = (§)*+(£)*'
where the brackets are inserted to indicate partial differential coefii-
cients. If we write this as
dp = (ji'f df + (j)\ du,
the normal to the tangent plane is evidently
and the equation of that plane
&(^-</))<^>'„=0.
293.] As a simple example, suppose a straight line to move along
a fixed straight line, remaining always perpendicular to it, while
rotating about it through an angle proportional to the space it has
advanced ; the equation of the ruled surface described will evidently
be p = at+u(PGOst + ysmt), fl)
where a, j8, y are rectangular vectorSj and
T0 = Ty.
This surface evidently intersects the right cylinder
p = a (p cos t + y sin t) + va,
in a helix (§§ 31 (m), 284) whose equation is
p = o^ + a(/3cos^ + ysini!).
These equations illustrate very well the remarks made in §§ 3 1 (^, 29 1
M
162 QUATERNIONS. [294.
as to the curves or surfaces represented by a vector equation ac-
cording as it contains one or two scalar variables.
From (1) we have
dp = \a—u{^svo.t—y<iOst)'\dt-\-{^ cos t-\-y&mt)du,
so that the normal at the extremity of p is
Ta {y cost-p sin t) - uT^^ Ua.
Hence, as we proceed along a generating line of the surface, for
which t is constant, we see that the direction, of the normal changes.
This, of coursCj proves that the surface is not developable.
294.] Hence the criterion for a developable surface is that if it
be expressed by an equation of the form
p = <f)t-\- tixj/t,
where (j)t and \jft are vector functions, we must have the direction of
the normal 'F{<t)'t + wft} \j/t
independent of u.
This requires either F-fi-^'t = 0,
which would reduce the surface to a cylinder^ all the generating
lines being parallel to each other ; or
F(j>'t\}/t = 0.
This is the criterion we seek, and it shews that we may write, for a
developable surface in general, the equation
p = (pt + U(p't • (1)
Evidently p = ^t
is a curve (generally tortuous) and (f/t is a tangent vector. Hence
a developable surface is the locus of all tangent lines to a tortuous
curve.
Of course the tangent plane to the surface is the osculating plane
at the corresponding point of the curve ; and this is indicated by
the fact that the normal to (1) is parallel to
r(i>tcl>"t. (See § 283.)
To find the form of the section of the surface made iy a normal plane
through a point in the curve.
The equation of the surface is
OT = p+«/3' + — p" + &c.+«(p'+*p'' + &c.).
The part of tsr— p which is parallel to p' is
-p'^(^-p)p'=-/(-(s+«^)-p"^(4+^) + ...);
therefore ^-p = Ap'+(~+ws) p"-(~ +'^) p'FpY' + ... .
297-] GEOMETEY OP CUBVES AND SURFACES. 163
And, when A = 0, i.e. in the normal section, we have approximately
so that ^ _ p = _ i_ p" _ 1_ p' fp'p'".
Z o
Hence the curve has an equation of the form
a semicubical parabola.
395. J A Geodetic line is a curve drawn on a surface so that its
osculating plane at any point contains the normal to the surface.
Hence, if v be the normal at the extremity of p, p and p" the first
and second differentials of the vector of the geodetic,
S.vp'p"= 0,
which may be easily transformed into
V.vdUp'^ 0.
296.] In the sphere Tp = ayte, have
V Up,
hence S.pp'p"= 0,
which shews of course that p is confined to a plane passing through
the origin, the centre of the sphere.
For a formal proof, we may proceed as follows —
The above equation is equivalent to the three
S9p = 0, Sdp'= 0, Sdp"= 0,
from which we see at once that 5 is a constant vector, and therefore
the first expression, which includes the others, is the complete in-
tegral.
Or we may proceed thus —
= -pS.ppY+p"s.py= r. Vpp'rpp"= r. Vpp'dVpp',
whence by § 133 (2) we have at once
UVpp'= const. = suppose,
which gives the same results as before.
297.] In any cone we have, of course,
Svp = 0,
since p lies in the tangent plane. But we have also
Svp'= 0.
Hence, by the general equation of § 295, eliminating v we get
= S.pp'rp'p"= SpdUp' by § 133 (2).
Integrating C=Sp Up'-jsdp Up'= Sp Up' +J Tdp.
The interpretation of this is, that the length of any arc of the geo-
detic is equal to the projection of the side of the cone (drawn to its
164 QTJATEENIONS. [298.
extremity) upon the tangent to the g«odetic. In other words, when
the cone is developed the geodetic becomes a straight line. A similar
result may easily be obtained for the geodetic lines on any develop-
able surface whatever.
298.] To find the shortest line connecting two points on a given
surface.
Here / Tdp is to be a minimum, subject to the condition that dp
lies in the given surface.
Now h^Tdp = fbTdp = -f^^^ = -fs. Udpdbp
= - [_S. Udp 8/)] + fs.bpdUdp,
where the term in brackets vanishes at the limits, as the extreme
points are fixed, and therefore 8p = 0.
Hence our only conditions are
P
' S.bpdUdp = 0, and Svbp = 0, giving
V.vdVdp = 0, as in § 295.
If the extremities of the curve are not given, but are to lie on
given curves, we must refer to the integrated portion of the ex-
pression for the variation of the length of the arc. And its form
S.Udpbp
shews that the shortest line cuts each of the given curves at right
angles.
299.] The osculating plane of the curve
p^^t
is S.4,'t<i,"t{m-p) = 0, (1)
and is, of course, the tangent plane to the surface
p = <t)t + U(t>'t (2)
Let us attempt the converse of the process we have, so far, pursued,
and endeavour to find (2) as the envelop of the variable plane (1).
Differentiating (1) with respect to t only, we have
By this equation, combined with (1), we have
^-p\\r.r<t>'rr<i>T\\<i^',
or zT = p + u(l)'= (l)+'U(l/,
which is equation (2).
300.] This leads us to the consideration of envelops generally,
and the process just employed may easily be extended to the problem
302.J GEOMETRY OF CURVES AND SURFACES. 165
of finditiff the envelop of a series of surfaces whose equation contains
one scalar parameter.
When the given equation is a scalar one^ the process of finding
the envelop is precisely the same as that employed in ordinary
Cartesian geometry, though the work is often shorter and simpler.
If the equation be given in the form
p =-\}i{t, u, v),
where t/^ is a vector function, t and u the scalar variables for any
one surface, v the scalar parameter, we have for a proximate surface
Pi = V' {h> %. ^i) = p+'Vt^t + 'Vu^'^'^Vv^'"-
Hence at all points on the intersection of two successive surfaces
of the series we have
which is equivalent to the following scalar equation connecting the
quantities t, u, and v ;
This equation, along with
p, = -f{i, u, v),
enables us to eliminate t, u, v, and the resulting scalar equation
is that of the required envelop.
301.] As an example, let us find the envelop of the osculating
plane of a tortuous curve. Here the equation of the plane is (§ 299),
S.{m-p)<i/t<i>"t= 0,
or CT = (l>t+x^'t+i/^"t = •^{x,y, {),
if p = <f)t
be the equation of the curve.
Our condition is, by last section,
or S.<i>'t 4>"t l(t)'t + so4>"t + y ^'"t] = 0,
or y84't<^"t<^"'t=(i.
Now the second factor cannot vanish, unless the given curve
be plane, so that we must have
and the envelop is 'si =■ <pt + w<^'t
the developable surface, of which the given curve is the edge of
regression, as in § 299.
302.] When the equation contains two scalar parameters its
differential coefiieients with respect to them must vanish, and we
have thus three equations from which to eliminate two numerical
quantities.
166 QUATERNIONS. [303.
A very common form in whieli these two parameters appear ia
quaternions is that of an unknown unit-vector. In this case the
problem may be thus stated — Find the envelop of the surface whose
scalar equation is Jpu^ a) = 0,
wJiere a is subject to the one condition
Ta = 1.
Differentiating with respect to o alone, we have
Svda = 0, Sada = 0,
where v is a known vector function of p and a. Since da may have
any of an infinite number of values, these equations shew that
Fav = 0.
This is equivalent to two scalar conditions only, and these, in addi-
tion to the two given scalar equations, enable us to eliminate a.
With the brief explanation we have given, and the examples
which follow, the student will easily see how to deal with any other
set of data he may meet with in a question of envelops.
303.] Find the envelop of a plane whose distance from the origin is
constant.
Here Sap =-—c,
with the condition Ta = 1 .
Hence, by last section, Vpa = 0,
and therefore p = ca,
or Tp = c,
the sphere of radius c, as was to be expected.
If we seek the envelop of those only of the planes which are parallel
to a given vector /3, we have the additional relation
Sa^ = 0.
In this case the three differentiated equations are
Spda = 0, Sada = 0, SjSda = 0,
and they give S.a^p = 0.
Hence a = U.^T^p,
and the envelop is TVfip = cTfi,
the circular cylinder of radius c and axis coinciding with fi.
By putting Safi = e, where e is a constant different from zero,
we pick out all the planes of the series which have a definite in-
clination to j8, and of course get as their envelop a right cone.
304.] The equation S'^ap+tS.a^p = h
represents a parabolic cylinder, whose generating lines are parallel
to the vector aFa/S. For the equation is of the second degree, and
305.] GEOMETRY OP CURVES AND SURFACES. 167
•
is not altered by increasing p by the vector xaFa^ ; also the surface
cuts planes perpendicular to a in one line, and planes perpendicular
to FajS in two parallel lines. Its form and position of course depend
upon the values of a, /3, and 6. It is required to find its envelop if ^
and b be constant, and a be subject to the one scalar condition
Ta=l.
The process of § 302 gives, by inspection,
pSap+ Vfip = oca.
Operating by S.a, we get
S^ap + S.aj3p =—«!,
which gives S.a/Sp = x-i- i.
But, by operating successively by S. Fj3p and by S.p, we have
{FPpf = (vS.aISp,
and {p^—x)Sap = 0.
Omitting, for the present, the factor Sap, these three equations give,
by elimination of x and a,
{rppf = p^{p^+b),
which is the equation of the envelop required.
This is evidently a surface of revolution of the fourth order whose
axis is /3 ; but, to get a clearer idea of its nature, put
and the equation becomes {V^taf = c* + 6zt^,
which is obviously a surface of revolution of the second degree,
referred to its centre. Hence the required envelop is the reciprocal
of such a surface, in the sense that t^e rectangle under the lengths of
condirectional radii of the two is constant.
We have a curious particular case if the constants are so related
that b + ^^ =zQ,
for then the envelop breaks up into the two equal spheres, touching
each other at the origin, P^ = ± ^^Pi
while the corresponding surface of the second order becomes the
two parallel planes S^.^ = + e^.
305.] The particular solution above met with, viz.
Sap — 0,
limits the original problem, which now becomes one of finding the
envelop of a line instead of a surface. In fact this equation, taken
in conjunction with that of the parabolic cylinder, belongs to that
generating line of the cylinder which is the locus of the vertices of
the principal parabolic sections.
168 QUATERNIONS. [306.
Our equations become 2S.al3p = h,
Sap = 0,
Ta = I;
whence Ffip = ica, giving
^^ 2
and thence ^^fip = - ',
so that the envelop is a circular cylinder whose axis is /3. [It is to
be remarked that the equations above require that
Sa^ = 0,
so that the problem now solved is merely that of tke envelop of a
parabolic cylinder which rotates about its focal line. This discussion
has been entered into merely for the sake of explaining a peculiarity
in a former result, because of course the present results can be
obtained immediately by an exceedingly simple process.]
306.] The equation SapS.ajip = a^,
with the condition Ta= I,
represents a series of hyperbolic cylinders. It is required to find
their envelop.
As before, we have pS.app+ F/SpSap = xa,
which by operating by S.a, S.p, and S. Vfip, gives
2a^ =—x,
p^S.afip = xSap,
{rfipySap=xS.afip.
Eliminating a and x we have, as the equation of the envelop,
p^iFjSpf = 4.a*.
Comparing this with the equations
p^=-2a^,
and {rppY = -2a^,
which represent a sphere and one of its circumscribing cylinders,
we see that, if eondirectional radii of the three surfaces be drawn
from the origin, that of the new surface is a geometric mean be-
tween those of the two others.
307.] Find the envelop of all spheres which touch one given line
and have their centres in another.
Let p = ^-\-yy
be the line touched by all the spheres, and let xa be the vector of
the centre of any one of them, the equation is (by § 200, or § 201)
y'^ip-xaf =-{r.y{fi-xa)Y,
3o8.] GEOMETEY OF CUEVES AND SURFACES. 169
•
ov, putting for simplicityj but without loss of generality,
Ty=l, Sa^ = 0, iSl3y = 0,
so that /3 is the least vector distance between the given lines,
{p—xa)^ = {^—xa)^-\-x'^S'^ay,
and, finally, P^-fi^- ix Sap = x^ S^ay.
Hence, by § 300, —2Sap = 2xS^ay.
[This gives no definite envelop if
Say = 0,
i. e. if the line of centres is perpendicular to the line touched by all
the spheres.]
Eliminating x, we have for the equation of the envelop
which denotes a surface of revolution of the second degree, whose
axis is a.
Since, from the form of the equation, Tp may have any magnitude
not less than T^, and since the section by the plane
Sap =
is a real circle, on the sphere
the surface is a hyperboloid of one sheet.
[It will be instructive to the student to find the signs of the
values of ^1,^2) ffs ^^ i^ § ^^^j ^^^ thence to prove the above con-
clusion.]
308.] As a final example let us find the envelop of the hyperbolic
cylinder SapS^p—o = 0,
where the vectors a and /3 are subject to the conditions
Ta = T^^ 1,
Say = 0, aS^8 = 0,
y and 6 being given vectors.
[It will be easily seen that two of the six scalars involved in a, /3
still remain as variable parameters.]
We have Sada = 0, Syda = 0,
so that da = xVay.
Similarly ^/3=yFj35.
But, by the equation of the cylinders,
SapSpd/S + SpdaSfip = 0,
or ySapS.^hp +xS.aypSfip = 0.
Now by the nature of the given equation, neither Sap nor S^p can
vanish, so that the independence of da and d^ requires
S.ayp = 0, S.fibp = 0.
170 QUATEE,]SriONS. [309-
Hence a = U.y Fyp, fi =U.h Ftp,
and the envelop is T.FypFbp — cTyb = 0,
a surface of the fourth order^ which may be constructed by laying
off mean proportionals between the lengths of condirectional radii
of two equal right cylinders whose axes meet in the origin.
309.] "We may now easily see the truth of the following general
statement.
Suppose the given equation of the series of surfaces, whose envelop
is required, to contain m vector, and n scalar, parameters ; and that
the latter are subject top vector, and q scalar, conditions.
In all there are 3m +n scalar parameters, subject to 3p + q scalar
conditions.
That there may be an envelop we must therefore in general have
{3m + n) — {3_p + q) = 1, or = 2.
In the former case the enveloping surface is given as the locus of a
series of curves, in the latter of a series ot points.
Differentiation of the equations gives us 3j) + q+l equations,
linear and homogeneous in the 3m+n differentials of the scalar
parameters, so that by the elimination of these we have one final
scalar equation in the first case, two in the second ; and thus in each
case we have just equations enough to eliminate all the arbitrary
parameters.
310.] To find the locus of the foot of the perpendicular drawn from
the origin to a tangent plane to any surface.
If Svdp =
be the differentiated equation of the surface, the equation of the
tangent plane is S(T!r — p)v=0.
We may introduce the condition
Svp = 1,
which in general alters the tensor of v, so that v~^ becomes the
required vector perpendicular, as it satisfies the equation
Smv = 1 .
It remains that we eliminate p between the equation of the given
surface, and the vector equation
The result is the scalar equation (in vr) required.
For example, if the given surface be the ellipsoid
^p4>P = 1.
we have ■sr"^ = v = 4>p,
3 1 3-] GEOMETRY OF CUEVES AND SURFACES. 171
•
so that the required equation is
or /Sar^-V = OT*,
which is Fresnel's Surface of Elasticity. (§ 263.)
It is well to remark that this equation is derived from that of the
reciprocal ellipsoid Sp(b-''-p = 1
by putting ot~^ for p.
3 11. J To find the reciprocal of a given surface with respect to the
unit sphere whose centre is the origin.
With the condition 8pv = 1,
of last section, we see that — u is the vector of the pole of the
tangent plane S{vT-p)v =().
Hence we must put zj=—v,
and eliminate phj the help of the equation of the given sm-faee.
Take the ellipsoid of last section, and we have
so that the reciprocal surface is represented by
It is obvious that the former ellipsoid can be reproduced from this
by a second application of the process.
And the property is general, for
Spv = 1
gives, by differentiation, and attention to the condition
Svdp = 0,
the new relation Spdv = 0,
so that p and r are corresponding vectors of the two surfaces : either
being that of the pole of a tangent plane drawn at the extremity of
the other.
312.] If the given surface be a cone with its vertex at the origin,
we have a peculiar case. For here every tangent plane passes
through the origin, and therefore the required locus is wbolly at an
infinite distance. The difficulty consists in Spv becoming in this
case a numerical multiple of the quantity which is equated to zero
in the equation of the cone, so that of course we cannot put as above
Spv = 1.
313.] The properties of the normal vector v enable us to write
the partial differentia] equations of families of surfaces in a very
simple- form.
Thus the distinguishing property of Cylinders is that all their
172 QUATERNIONS. ' [SH-
generating lines are parallel. Hence all positions of v must be
parallel to a given plane — or
Sav = 0,
which is the quaternion form of the well-known equation
,dF dF dF „
dx dy dz
To integrate it, remember that we have always
Svdp = 0,
and that as v is perpendicular to a it may be expressed in terms of
any two vectors, /3 and y, each perpendicular to a.
Hence v = x^ + yy,
and xS^dp + ySydp = 0.
This shews that S^p and Syp are together constant or together
variable, so that SfSp =f{Syp),
where/" is any scalar function whatever.
314.] In Surfaces of Bevolution the normal intersects the axis.
Hence, taking the origin in the axis a, we have
S.apv = 0,
or V = xa + yp.
Hence xSadp + ySpdp = 0,
whence the integral Tp =f{Sap).
The more common form, which is easily derived from that just
written, is TFap = F{Sap).
In Cones we have Svp = 0,
and therefore
Svdp = S.v{TpdUp+ UpdTp) = TpSvdUp.
Hence SvdUp = 0,
so that V must be a function of Up, and therefore the integral is
AUp) = 0,
which simply expresses the fact that the equation does not involve
the tensor of p, i. e. that in Cartesian coordinates it is homogeneous.
315.] If equal lengths he laid off on the normals drawn to any
surface, the new surface formed hy their extremities is normal to the
same lines.
For we have w = p + a Uv,
and SvdTn = Svdp + aSvdUv = 0,
which proves the proposition.
Take, for example, the surface
Sp(l>p = 1 ;
3 1 7-] GEOMETRY OP CURVES AND SURFACES. 173
the above equation becomes
so that ^'=(^ +
and the equation of the new surface is to be found by eliminating
~— (written ») between the equations
1 = <S'.(«(j!)+l)-i,!r<^(a;0+l)-iOT,
-1
a
and i=S4 (xcj) + 1 )-^zj(j) {xs^ + 1 y-'^-nr.
316.] It appears from last section that if one orthogonal surface
can be drawn cutting a given system of straight lines, an inde-
finitely great number may be drawn: and that the portions of
these lines intercepted between any two selected surfaces of the
series are all equal.
Let p = a+XT,
where o- and t are vector functions of p, and x is any scalar, be the
general equation of a system of lines : we have
Srdp = = S{p—a)dp
as the differentiated equation of the series of orthogonal surfaces, if
it exist. Hence the following problem.
317.] It is required to find the criterion of integrahility of the
equation Svdp = (1)
as the complete differential of the equation of a series of mrfaces.
Hamilton has given [Elements, p. 702) an extremely elegant solu-
tion of this problem, by means of the properties of linear and vector
functions. We adopt a different and somewhat less rapid process,
on account of some results it offers which will be useful to us in
the next Chapter ; and also because it will shew the student the
connection of our methods with those of ordinary differential equa-
. tions.
If we assume Fp= C
to be the integral, and apply to it the very singular operator de-
Adsed by Hamilton,
„ . d . d , d
dx '' dy dz
^ .dF .dF ,dF
we have vi^= .^ +^^ +^^-
174 QUATERNIONS. [3 1 8.
But p = ix +jy + hz,
whence dp ■= idx+jdy-\-kdz,
,^ dF , dF , dF , „,„T,
and Q = dF=-rdx-\--^dy-ir^rdz——SdpVF.
dx dy ^ dz '^
Comparing with the given equation, we see that the latter repre-
sents a series of surfaces if p, or a scalar multiple of it, can be ex-
pressed as VF.
If v = VF,
„,-^ ^d^F d'^F d^Fs
we have ^^ = V^^=-(^ + ^ + ^) '
a well-known and most important expression, to which we shall
return in next Chapter. Meanwhile we need only remark that
the last-written quantities are necessarily scalars, so that the only
requisite condition of the integrability of (1) is
rVv= (2)
If V do not satisfy this criterion, it may when multiplied by a scalar.
Hence the farther condition
rv (wv) = 0,
which may be written
FvVw—wrVv = (3)
This requires that SvVv = (4)
If then (2) be not satisfied, we must try (4). If (4) be satisfied to
will be found from (3) ; and in either case (1) is at once integrable.
[If we put dv = (t>dp
where </> is a linear and vector function, not necessarily self-con-
jugate, we have
rvv=:r(i^ + ...) = rii<t,i+...)=-e,
by § 173. Thus, if (j) be self-conjugate, e = 0, and the criterion (2)
is satisfied. If (j) be not self-conjugate we have by (4) for the cri-
terion Sev= 0.
These results accord with Hamilton's, lately referred to, but the
mode of obtaining them is quite difierent from his.]
3I8.3 As a simple example let us first take lines diverging from, a
point. Here v\[p, and we see that \i v = p
Vz; = -3,
so that (2) is satisfied. And the equation is
Spdp = 0,
whose integral Tp ■=■ C
gives a series of concentric spheres.
3 1 9-] GEOMETEY OF CUEVES AND SUEPACBS. 175
Lines ^perpendicular to, and intersecting, it, fixed line.
If a be the fixed line, ^ any of the others, we have
S.a^p = 0, Sa^ =Q, Spdp = 0.
Here i- \\ aVap,
and therefore equal to it, because (2) is satisfied.
Hence S.dpaVap = 0,
or S.VapFadp = 0,
whose integral is the equation of a series of right cylinders
T^rap= C.
319.] To find the orthogonal trajectories of a series of circles whose
centres are in, and their planes perpendicular to, a given line.
Let a be a unit-vector in the direction of the line, then one of
the circles has the equations
Tp = G,\
Sap = C, 3
where G and C are any constant scalars whatever.
Hence, for the required surfaces
V II d^p II Fap,
where d^p is an element of one of the circles, v the normal to the
orthogonal surface. Now let dp be an element of a tangent to the
orthogonal surface, and we have
Svdp — S.apdp = 0.
This shews that dp is in the same plane as a and p, i.e. that the
orthogonal surfaces are planes passing through the common axis.
[To integrate the equation S.apdp =
evidently requires, by § 317^ the introduction of a factor. For
rvFap = riirai+jVaj + Wak)
= 2a,
so that the first criterion is not satisfied. But
S.FaprVFap = 2S.arap = 0,
so that the second criterion holds. It gives, by (3) of § 317,
F.Vu;Fap+2wa = 0,
or pSaVto — aSpVw + 2 wa = 0.
That is SaVw = 0, \
SpVw = 2w. J
These equations are satisfied by
But a simpler mode of integration is easily seen. Our equation
may be written
= S.aF^ = Sa^-fi = ^.^alog^Z-^
p Up p
176 QUATERNIONS. [320.
which is immediately integrable, j3 being an arbitrary but constant
vector.
As we have not introduced into this work the logarithms of ver-
sors, nor the corresponding angles of quaternions, we must refer to
Hamilton's Elements for a farther development of this point.]
320.] To jmd the orthogonal trajectories of a given series of sur-
faces.
If the equation Fp = C,
give Svdp — f^,
the equation of the orthogonal curves is
Vvdp = 0.
This is equivalent to two scalar differential equations (§ 197), which,
when the problem is possible, belong to surfaces on each of which
the required lines lie. The finding of the requisite criterion we
leave to the student.
Let the surfaces be concentric spheres.
Here p^ ^ g.
and therefore Vpdp= 0.
Hence Tp^ dUp=-Up Fpdp = 0,
and the integral is Up = constant,
denoting straight lines through the origin.
Let the sv/rfaces be spheres touching each other at a common point.
The equation is (§ 2 1 8)
Sap-^ = G,
whence V.papdp = 0.
The integrals may be written
S.aPp = 0, p^+hTVap = 0,
the first (/3 being any vector) is a plane through the common dia-
meter; the second represents a series of rings or tores (§323) formed
by the revolution, about a, of circles touching that ILae at the point
common to the spheres.
Let the surfaces be similar, similarly situated, and concentric, sur-
faces of the second order.
Here Spxp = C,
therefore ^XP^P = ^•
But, by § 290, the integral of this equation is
p = e'^e
where (f> and x are related to each other, as in § 290 ; and e is any
constant vector.
321.] GEOMETEY OF CURVES AND SUEPACES. 177
331 .J To integrate the linear partial differential equation of a
family of surfaces.
The equation (see § 3 1 3)
dx dy dz ~
may be put in the very simple form
S (o-V) V, ■= 0,
if we write a- = iP+JQ + kB,
1 „ . d . d , d
and V=t- — |-;---j.^--.
dx '' dy dz
This gives, at once, Vu = t/iFOcr,
where »« is a scalar and 6 a vector (in whose tensor m might have
been included, but is kept separate for a special purpose). Hence
dit = — S{dpV)ii
= —mS.dddp
= —S.edr,
if we put dT = mr.<Tdp
so that m is an integrating factor of V. (rdp. If a value of m can be
found, it is obvious, from the form of the above equation, that d
must be a function of r alone ; and the integral is therefore
w = F{t) = const,
where F is an arbitrary scalar function.
Thus the differential equation of Cylinders is
' S(dV)u=0,
where a is a constant vector. Here m=l, and
M = F{Fap) = const.
That of Cones referred to the vertex is
S{pV)u= 0.
Here the expression to be made integrable is
r.pdp.
But Hamilton long ago shewed that (§133 (2))
dUp _ ydp _ V.pdp
-W~ P~ {Tpf '
which indicates the value of m, and gives
u = F{TJp) — const.
It is obvious that the above is only one of a great number of
different processes which may be applied to integrate the differential
equation. It is quite easy, for instance, to pass from it to the
assumption of a vector integrating factor instead of the scalar m,
N
178 QUATERNIONS. [322.
aud to derive tlie usual criterion of integrability. There is no diffi-
culty in modifying the process to suit the case when the right-hand
member is a multiple of u. In fact it seems to throw a very clear
light upon the whole subject of the integration of partial differ-
ential equations. If, instead of S (o-V), we employ other operators
as S {(tV) S {tV), S.o-VtV, &c. (where V may or may not operate on
u alone), we can pass to linear partial differential equations of the
second and higher orders. Similar theorems can be obtained from
vector operations, as V{<tV)*.
322.] Find the general equation of surfaces described by a line
which always meets, at right angles, a fixed line.
If a be the fixed line, y3 and y forming with it a rectangular unit
system, then p = a;a +y + zy),
where y may have all values, but x and z are mutually dependent,
is one form of the equation.
Another, expressing the arbitrary relation between x and z is
But we may also write
p = aF{x) +ya''P,
as it obviously expresses the same conditions.
The simplest case is when F{x) = hx. The surface is one which
cuts, in a right helix, every cylinder which has a for its axis.
323.] The centre of a sphere moves in a given circle, find the equa-
tion of the ring described.
Let a be the unit-vector axis of the circle, its centre the origin,
r its radius, a that of the sphere.
Then [p-^f =-0^
is the equation of the sphere in any position, where
<So/3 = 0, 2)3 = n
These give S.a^p = 0, and ^ must now be eliminated. The result
is that ^ = raUVap,
giving (p^— r^-t-a^)^ = ^r^T'^Vap,
= 4r^-p^-S^ap),
which is the required equation. It may easily be changed to
(p^-a'^ + r^)^ =-4:a^p^-4:rWap, ...: (1)
and in this form it enables us to give an immediate proof of the
very singular property of the ring (or tore) discovered by Villarceau.
* Tait, Proc. R. S. E., 1869-70.
324.J GEOMETRY OF CURVES AND SURFACES. 179
For the planes S.p (a± ) = 0,
which together are represented by
r^{r^-a^)8^ap-a'^S^^p = 0,
evidently pass through the origin and touch (and cut) the ring.
The latter equation may be written
r'^S^ap-a^{8^ap + S^pU^) = 0,
or r^S^ap + a^{p^ + S^.apU^) =0 (2)
The plane intersections of (1) and (2) lie obviously on the new
surface (^2_^2 + y2)2 ^ ia^S^.apUia,
which consists of two spheres of radius r, as we see by writing its
separate factors in the form
(p±aaUpf+r^ = 0.
334.] It may be instructive to work out this problem from a
different point of view, especially as it affords excellent practice in
transformations.
A circle revolves about an axis passing within it, the perpendicular
from the centre on the axis lying in the plane of the circle: shew that,
for a certain position of the axis, the same solid mny he traced out by a
circle revolving about an external axis in its own plane.
Let a = •fh'^ + c^ be the radius of the circle, i the vector axis of
rotation, —ca (where Ta=-\) the vector perpendicular from the
centre on the axis i, and let the vector
hi + da
be perpendicular to the plane of the circle.
The equations of the circle are
(p_ca)2 + ^2 + c2 = 0, \
S(i + Yia)p = 0. C
Also —p^ = S^ip + S^ap + S'^.iap,
b^
= SHp + S^ap+ -^SHp
by the second of the equations of the circle. But, by the first,
(/)2 + 5Z)2 = 4c2/SV = -4 {c^p'^+a^SHp),
which is easily transformed into
{(?-¥f=-i.a^{p^ + S^ip),
or p2_52 ^ —2aTrip.
If we put this in the forms
p^-h^ = 2aSpp,
and {p-a^f + c^=:0,
N a
180 QUATERNIONS. [32 5-
where ;3 is a unit-vector perpendicular to i and in tlie plane of i
and p, we see at once that the surface will be traced out by a circle
of radius c, revolving about i, an axis in its own plane^ distant a
from its centre.
This problem is not well adapted to shew the gain in brevity and
distinctness which generally follows the use of quaternions ; as,
from its very nature, it hints at the adoption of rectangular axes
and scalar equations for its treatment, so that the solution we have
given is but little different from an ordinary Cartesian one.
325.] A surface is generated hy a straight line which intersects two
fixed lines : find the general equation.
If the given lines intersect, there is no surface but the plane con-
taining them.
Let then their equations be,
p = a + xfi, p = a^ + XiPi-
Hence every point of the surface satisfies the condition, § 30,
p=y(a + a;^) + (l-5^)(ai + 3'i^i) (1)
Obviously y may have any value whatever : so that to specify a
particular surface we must have a relation between x and x^. By
the help of this, x^ may be eliminated from (1), which then takes
the usual form of the equation of a surface
P = 't>i'«,^)-
Or we may operate on (1) by F.(a + xj3-- ai—XiJ3i), so that we get
a vector equation equivalent to two scalar equations (§§ 98, 116),
and not containing y. From this x and x^ may easily be found in
terms of p, and the general equation of the possible surfaces may be
written /"{^t *i) = 0,
where /" is an arbitrary scalar function, and the values of x and x^
are expressed in terms of p.
This process is obviously applicable if we have, instead of two
straight lines, any two given curves through which the line must '
pass ; and even when the tracing line is itself a given curve, situated
in a given manner. But an example or two will make the whole
process clear.
326.] Suppose the moveable line to le restricted by the condition
that it is always parallel to a fixed plane.
Then, in addition to (1), we have the condition
Sy{a-i^-\-x-yP-^—a — x^) = 0,
y being a vector perpendicular to the fixed plane.
We lose no generality by assuming o and Oj, which are any
327.] GKOMETEY OF CURVES AND SUEFACES. 181
vectors drawn from the origin to the fixed lines, to be each per-
pendicular to y ; for, if for instance we could not assume Sya = 0, it
would follow that Sy^ = 0, and the required surface would either
be impossible, or would be a plane, cases which we need not con-
sider. Hence x^8y^^-x8y^ - 0.
Eliminating' ajj, by the help of this equation, from (1) of last section,
we have , „; , , ^ Sy& -.
Operating by any three non-coplanar vectors and with the charac-
teristic S, we obtain three equations from which to eliminate a; and y.
Operating by S.y we find
Syp = xSjSy.
Eliminating x by means of this, we have finally
^■'(« + ^^)(«.+ ^) = «.
which appears to be of the third order. It is really, however, only
of the second order, since, in consequence of our assumptions, we
have Vauj^ \\ y,
and therefore Syp is a spurious factor of the left-hand side.
327.] Let the fixed lines he perpendicular to each other, and let
the moveable line pass through the circumference of a circle, whose
centre is in the common perpendicular, and whose plane bisects that line
at right angles.
Here the equations of the fixed lines may be written
p = a + x^, p =— a+a?iy,
where a, j3, y, form a rectangular system, and we may assume the
two latter to be unit-vectors.
The circle has the equations
p^ =—a^, Sap = 0.
Equation (1) of § 325 becomes
p = i/{a+xj3} + {l-if){-a + x^y).
Hence Sar'^p = y—(l—^] = 0, or y = i-
Also p2= -«2 = (2y-l)2 a'-x^f-xl (1-^)^
or 4fl^ = (x^+xl),
so that if we now suppose the tensors of /3 and y to be each 2 a, we
may put x = cos 0, x^ = sin 6, from which
p = (2j^— l)a + y/3cos0+(l— y)ysin5;
^•^^ ^^""^ {l+Sa-^pf + {l-Sa-^pf = '^ •
182 QUATBRNIONS. [328.
For this very simple case the solution is not better than the
ordinary Cartesian one; but the student will easily see that we
may by very slight changes adapt the above to data far less sym-
metrical than those from which we started. Suppose, for instance,
/3 and y not to be at right angles to one another ; and suppose the
plane of the circle not to be parallel to their plane, &c., &c. But
farther, operate on every line in space by the linear and vector
function (^, and we distort the circle into an ellipse, the straight
lines remaining straight. If we choose a form of ^ whose principal
axes are parallel to a, p, y, the data will remain symmetrical, but
not unless. This subject will be considered again in- the next
Chapter.
328.] To find the curvature of a normal section of a central surface
of the second order.
In this, and the few similar investigations which follow, it will
be simpler to employ infinitesimals than differentials ; though for a
thorough treatment of the subject the latter method, as may be seen
in Hamilton's Elements, is preferable.
We have, of course, '^/'</>P = Ij
and, if p + hp be also a vector of the surface, we have rigorously,
whatever be the tensor ofbp,
Sip + 8p)<t>{p + bp)= 1.
Hence 2Sbpcl)p-\-Sbp<j)bp = (1)
Now </)p is normal to the tangent plane at the extremity of p, so
that if t denote the distance of the point p + bp from that plane
i =-SbpU(l)p,
and (1) may therefore be written
•itT<i>p-T^bpS.mp^Ubp = 0.
But the curvature of thfe section is evidently
"^ T^bp '
or, by the last equation,
±-^^s.mp<i>mp.
In the limit, bp is a vector in the tangent plane ; let ct- be the vector
semidiameter of the surface which is parallel to it, and the equation
of the surface gives T^isS .U-stcjjU-st = 1,
so that the curvature of the normal section, at the point p, in the
direction of or, is 1
329-J GEOMETRY OP CURVES AND SURFACES. 183
•
Hirectly as the perpendicular from the centre on the tangent plane, and
inversely as the square of the semidiameter parallel to the tangent line,
a well-known theorem.
329.] By the help of the known properties of the central section
parallel to the tangent plane, this theorem gives us all the ordinary
properties of the directions of maximum and minimum curvature,
their being at right angles to each other, the curvature in any
normal section in terms of the chief curvatures and the inclination
to their planes, &c., &c., without farther analysis. And when, in a
future section, we shew how to find an osculating surface of the
second order at any point of a given surface, the same properties
will be at once established for surfaces in general. Meanwhile we
may prove another curious property of the surfaces of the second
order, which similar reasoning extends to all surfaces.
The equation of the normal at the point p + 8p in the surface
treated in last section is
CT- = /3 + 8p+«(^(p + 8/)) (1)
This intersects the normal at p if (§§ 203, 210)
S.hp^p^hp = 0,
that is, by the result of § 273, if 8p be parallel to the maximum or
minimum diameter of the central section parallel to the tangent
plane.
Let o-j and o-g be those diameters, then we may write in general
hp =piTi + q(T2,
where ^ and q are scalars, infinitely small.
If we draw through a point P in the normal at p a line parallel
to (Tj, we may write its equation
OT = p-{-a(j)p+^a^.
The proximate normal (1) passes this line at a distance (see § 203)
S . {a(l>p — bp) UF(Ti (t){p + 8/)),
or, neglecting terms of the second order,
,,,-p- ■ (op 84pu-i(i)iT-^ + aqS.(l)p<jj(p<T2 + q S.cria^fjyp).
IT (r-j(pp
The first term in the bracket vanishes because o-j is a principal vector
of the section parallel to the tangent plane, and thus the expression
becomes / a „ \
Hence, if we take a — Tel, ^^ distance of the normal from the new
line is of the second order only. This makes the distance of P from
the point of contact T(f>pT(Tl, i.e. the principal radius of curvature
184 QUATERNIONS. [330.
along the tangent line parallel to o-g. That is, the group of normals
drawn near a point of a central surface of the second order pass ulti-
mately through two lines each parallel to the tangent to one principal
section, and passing through the centre of curvature of the other. The
student may form a notion of the nature of this proposition by con-
sidering a small square plate, with normals dravra at every point,
to he slightly bent, but by different amounts, in planes perpendicular
to its edges. The first bending will make all the normals pass
through the axis of the cylinder of which the plate now forms part ;
the second bending will not sensibly disturb this arrangement,
except by lengthening or shortening the line in which the normals
meet, but it will make them meet also in the axis of the new
cylinder, at right angles to the first. A small pencil of light, with
its focal lines, presents this appearance, due to the fact that a series
of rays originally normal to a surface remain normals to a surface
after any number of reflections and refractions. (See § 315).
330.] To extend these theorems to surfaces in general, it is only
necessary, as Hamilton has shewn, to prove that if we write
dv = (\)dp,
is a self-conjugate function ; and then the properties of <|), as ex-
plained in preceding Chapters, are applicable to the question.
As the reader will easily see^ this is merely another form of the
investigation contained in § 317. But it is given here to shew
what a number of very simple demonstrations may be given of
almost all quaternion theorems.
The vector v is defined by an equation of the form
dfp = Svdp,
where /" is a scalar function. Operating on this by another inde-
pendent symbol of differentiation, 8, we have
hdfp = Sbvdp + Svhdp.
In the same way we have
dbfp = Sdvhp + Svdbp.
But, as d and 8 are independent, the left-hand members of these
equations, as well as the second terms on the right (if these exist
at all), are equal, so that we have
Sdvbp = Shvdp.
This becomes, putting dv = <^dp,
and therefore Sv = ^6p,
8bp<pdp = Sdptjibp,
which proves the proposition.
333-] GEOMETRY OP CURVES AND SURFACES. 185
331.] If we write the differential of the equation of a surface in
the form df(t = iSvAp,
then it is easy to see that
f{p-\-dp) =fp+2Svdp + Sdvdp + kc.,
the remaining terms containing as factors the third and higher
powers of Tdp. To the second order, then, we may write, except
for certain singular points,
= 2Svdp + Sdvdp,
and, as before, (§ 328), the curvature of the normal section whose
tangent line is dp is 1 „ dv
Yv Tp'
333.] The step taken in last section, although a very simple one,
virtually implies that the first three terms of the expansion of
/(p + dp) are to be formed in accordance with Taylor's Theorem,
whose applicability to the expansion of scalar functions of quater-
nions has not been proved in this work, (see § 135); we therefore
give another investigation of the curvature of a normal section,
employing for that purpose the formulae of § (282).
We have, treating dp as an element of a curve,
Svdp = 0,
or, making s the independent variable,
Svp'= 0.
From this, by a second dififerentiation,
8^p' + Svp"= 0.
The curvature is, therefore, since v \\ p" and Tp'— \,
333.] Since we have shewn that
dv ^ (f)dp
where is a self-conjugate linear and vector function, whose con-
stants depend only upon the nature of the surface, and the position
of the point of contact of the tangent plane ; so long as we do not
alter these we must consider if) as possessing the properties explained
in Chapter V.
Hence, as the expression for Tp" does not involve the tensor of
dp, we may put for dp any unit-vector r, subject of course to the
condition Svt = 0. , (1)
And the curvature of the normal section whose tangent is r is
186 QUATERNIONS. [334-
If we consider the central section of the surface of the second order
&ss^^-\-Tv = 0,
made by the plane Svm = 0,
we see at once that the curvature of the given surface along the normal
section touched hy t is inversely as the square of the parallel radius in
the auxiliary surface. This, of course, includes Euler's and other
well-known Theorems.
334.J To find the directions of maximum and minimum curvature,
we have St<^t = max. or min.
with the conditions^ Svt = 0,
Tt= 1.
By differentiationj as in § 273, we obtain the farther equation
S.VT(\)T = (1)
If T be one of the two required directions, t'=tUv is the other, for
the last-written equation may be put in the form
S.TUv(t>{vTUv) = 0,
i.e. S.T'<t>{vT') = 0,
or 8.v/^T = 0.
Hence the sections of greatest and least curvature are perpendicular to
one another.
We easily obtain, as in § 273, the following equation
S.v{(f)-\-ST^T)-'^V = 0,
whose roots divided by Tv are the required curvatures.
335.] Before leaving this very brief introduction to a subject, an
exhaustive treatment of which will be found in Hamilton's Elements,
we may make a remark on equation (1) of last section
S.VT(i)T = 0,
or, as it may be written, by returning to the no'tation of § 333,
S.vdpdv = 0.
This is the general equation of lines of curvature. For, if we define
a line of curvature on any surface as a line such that normals drawn
at contiguous points in it intersect, then, bp being an element of
such a line, the normals
■ST = p + xv and ■or = p + 5p + y (v + bv)
must intersect. This gives, by § 203, the condition
, S.bpvbv = 0,
as above.
EXAMPLES TO CHAPTER IX. 187
EXAMPLES TO CHAPTER IX.
1 . Find the length of any arc of a curve drawn on a sphere so as
to make a constant angle with a fixed diameter.
2. Shew that, if the normal plane of a curve always contains a
fixed line, the curve is a circle.
3. Mnd the radius of spherica,l curvature of the curve
p = (jit.
Also find the equation of the locus of the centre of spherical
curvature.
4. (Hamilton, Bishop Law^ s Premium Examination, 1854.)
(a.) If p be the variable vector of a curve in space, and if the
differential Ak be treated as = 0, then the equation
dT{p-K) =
obliges K to be the vector of some point in the normal
plane to the curve.
(b.) In like manner the system of two equations, where dK
and d^K are each = 0,
dT(p-K) = 0, d^T{p-K) = 0,
represents the axis of the element, or the right line
drawn through the centre of the osculating circle, per-
pendicular to the osculating plane.
(c.) The system of the three equations, in which k is treated
as constant,
dT{p-K) = 0, d^T(p-K) = 0, d^T{p-K) = 0,
determines the vector k of the centre of the osculating
sphere.
{d.) For the three last equations we may substitute the follow-
ing :
S.{p—K)dp = 0,
S.{p-K)d\ + dp^ = 0,
S.{p-K)d^p + 3S.dpd^p = 0.
(e.) Hence, generally, whatever the independent and scalar
variable may be, on which the variable vector p of the
curve depends, the vector k of the centre of the oscu-
lating sphere admits of being thus expressed :
3 F.dpd^pS.dpd^p-dp^ F.dpd^p
K = p +
S.dpd^pd^p
188 QUATEEKIONS.
(/".) In general,
d{d-W.dpUp) = d{Tp-^r.pdp)
= Tp-'^ (sr.pdpS.pdp-p^r.pd^p) ;
whence,
^r.pdpS.pdp-pW.pd^P = p^Tpd{p-^F.dpUp);
and, therefore, the recent expression for k admits of
being thus transformed,
dp*d(dp-^r.d^pUdp )
"-P'^ S.d^pd^pUdp
iff.) If the length of the element of the curve be constant,
dTdp=0, this last expression for the vector of the centre
of the osculating sphere to a curve of double curva-
ture becomes, more simply^
d.d^pdp^
K = p +
or K = p +
S.dpd^pd^p '
F.d^pdp^
S.dpd^pd^p
{h.) Verify that this expression gives /c = 0, for a curve de-
scribed on a sphere which has its centre at the origin
of vectors ; or shew that whenever dTp = 0, d^Tp = 0,
d^Tp = 0, as well as dTdp = 0, then
pS.dp-''d''pd^p=r.dpdy.
5. Find the curve from every point of which three given spheres
ajjpear of equal magnitude.
6. Shew that the locus of a point, the difference of whose dis-
tances from each two of three given points is constant, is a plane
curve.
7. Find the equation of the curve which cuts at a given angle
all the sides of a cone of the second order.
Find the length of any are of this curve in terms of the distances
of its extremities from the vertex.
8. Why is the centre of spherical curvature, of a curve described
on a sphere, not necessarily the centre of the sphere ?
9. Find the equation of the developable surface whose generating
lines are the intersections of successive normal planes to a given
tortuous curve.
1 0. Find the length of an arc of a tortuous curve whose normal
planes are equidistant from the origin.
11. The reciprocals of the perpendiculars from the origin on the
tangent planes to a developable surface are vectors of a tortuous
EXAMPLES TO CHAPTER IX. 189
curve ; from whose osculatin^planes the cusp-edge of the original
surface may be reproduced by the same process.
12. The equation p=Fa'p,
where a is a unit- vector not perpendicular to ft represents an ellipse.
If we put y = Fa^, shew that the equations of the locus of the
centre of curvature are
S.pyp = 0,
Sipp + S^yp = {fiSUapf.
13. Find the radius of absolute curvature of a spherical conic.
14. If a cone be cut in a circle by a plane perpendicular to a side,
the axis of the right cone which osculates it, along that side, passes
through the centre of the section.
15. Shew how to find the vector of an umbilicus. Apply your
method to the surfaces whose equations are
Spipp = 1,
and SapS^pSyp = 1.
16. Find the locus of the umbilici of the surfaces represented by
the equation Sp {(p + A)-^p=l,
where A is an arbitrary parameter.
17. Shew how to find the equation of a tangent plane which
touches a surface along a line^ straight or curved. Find such planes
for the following- surfaces
Spipp = 1,
Sp{<j>-p^)-^p=l,
and {p^-a'^ + b^y + 4:{a^p^ + 6^S'^ap)= 0.
18. Find the condition that the equation
S{p + a)<l>p= 1,
where ^ is a self-conjugate linear and vector function, may represent
a cone.
19. Shew from the general equation that cones and cylinders are
the only developable surfaces of the second order.
20. Find, the equation of the envelop of planes drawn at each
point of an ellipsoid perpendicular to the radius vector from the
centre.
21. Find the equation of the envelop of spheres whose centres lie
on a given sphere, and which pass through a given point.
22. Find the locus of the foot of the perpendicular from the
centre to the tangent plane of a hyperboloid of one, or of two,
sheets.
190 QUATEENIONS.
23. 'H.arailtou, Mskqp Law's Premium Hxamination, 1852,
{a.) If p be the vector of a curve in space, the 'length of the
element of that curve is Tdp ; and the variation of the
length of a finite arc of the curve is
b/Tdp = -fSUdpbdp =-ASUdpbp+/SdUdpbp.
(5.) Hence, if the curve be a shortest line on a given surface,
for which the normal vector is v, so that Svbp = 0, this
shortest or geodetic curve must satisfy the differential
equation, FvdUdp = 0.
Also, for the extremities of the arc, we have the limiting
equations,
SUdpo Spo = J SUdp^ 8pi = 0.
Interpret these results,
(c.) For a spheric surface, Fvp = 0, pdUdp=Q ; the integrated
equation of the geodetics is p Udp = ■nr, giving Sxsp =
(great circle).
For an arbitrary cylindric surface,
Sav = 0, adUdp = ;
the integral shews that the geodetic is generally a helix,
making a constant angle with the generating lines of
the cylinder.
[d.) For an arbitrary conic surface,
Svp = 0, SpdUdp = ;
integrate this differential equation, so as to deduce from
it, TVpUdp = const.
Interpret this result ; shew that the perpendicular from
the vertex of the cone on the tangent to a given geo-
detic line is constant ; this gives the rectilinear develop-
ment.
When the cone is of the second degree, the same property
is a particular case of a theorem respecting confocal
surfaces,
(e.) For a surface of revolution,
S.apv — 0, S.apdUdp = ;
integration gives,
const. = S.apUdp = TVapSU (Fap.dp) ;
the perpendicular distance of a point on a geodetic
line from the axis of revolution varies inversely as the
cosine of the angle under which the geodetic crosses a
parallel (or circle) on the surface.
EXAMPLES TO CHAPTEE IX. 191
(/'.) The diiferential eqrration, S.apdUdp = 0, is satisfied not
only by the geodeties, but also by the circles, on a
surface of revolution ; give the explanation of this fact
of calculationj and shew that it arises from the coinci-
dence between the normal plane to the circle and the
plane of the meridian of the surface.
(g.) For any arbitrary surface, the equation of the geodetic
may be thus transformed, S.vdpcPp = ; deduce this
form, and shew that it expresses the normal property
of the osculating plane.
(A.) If the element of the geodetic be constant, dTdp = 0, then
the general equation formerly assigned may be reduced
to r.vd^p= 0.
Under the same condition, d^p = —v'^Sdvdp.
{i.) If the equation of a central surface of the second order
be put under the form fp = I, where the function _/
is scalar, and homogeneous of the second dimension,
then the diiferential of that function is of the forni
dfp = 2S.vdp, where the normal vector, v = <l>p, is a dis-
tributive function of p (homogeneous of the first dimen-
sion), dv=d(j)p = <l)dp.
This normal vector v may be called the vector of proximity
(namely, of the element of the surface to the centre) ;
because its reciprocal, v~^, represents in length and in
direction the perpendicular let fall from the centre on
the tangent plane to the surface.
(^.) If we make S<T<^p =y(o-, p), this function/" is commutative
with respect to the tvjo vectors on which it depends,
f{p, a) =/'(*, p) ; it is also connected with the former
functiony, of a single vector p, by the relation,/" (p, p) ■=fp :
so that fp = Sp<pp.
fdp = Sdpdv ; dfdp = 2S.dv d^p ; for a geodetic, with, con-
stant element,
2jdp V
this equation is immediately integrable, and gives
const. =Tv-J{fJJdp) = reciprocal of Joachimstal's pro-
duct, PB.
(l.) If we give the name of " Didonia" to the curve (discussed
by Delaunay) which, on a given surface and with a
given perimeter, contains the greatest area, then for
192 QUATERNIONS.
such a Didonian curve we have by quaternions the
formula, fS. Uvdpbp + c h/Tdp = 0,
where c is an arbitrary constant.
Derive hence the differential equation of the second order,
equivalent (through the constant c) to one of the third
order, g-^Sp = F. UvdUdp.
Geodeties are, therefore, that limiting case of Didonias for
which the constant c is infinite.
On a plane, the Didonia is a circle, of which the equation,
obtained by integration from the general form, is
p = ■uT + cUvdp,
m being vector of centre, and c being radius of circle.
(m.) Operating by 8. TJdp, the general differential equation of
the Didonia takes easily the following forms :
c'-'Tdp =S{Uvdp.dUdp);
c-^Tdp^ = S{Uvdp.d^p);
c-'^Tdp^ = S.Uvdpd^p;
Uvdp
{n.) The vector w, of the centre of the osculating circle to a
curve in space, of which the element Tdp is constant,
has for expression,
dp''
Hence for the general Didonia,
c"i = 5i
Uvdp
T{p-<^) = cSU'"
vdp
(o.) Hence, the radius of curvature of any one Didonia varies,
in general, proportionally to the cosine of the inclination
of the osculating plane of the curve to the tangent
plane of the surface.
And hence, by Meusnier's theorem, the difference of the
squares of the curvatures of curve and surface is con-
stant J the curvature of the surface meaning here the
reciprocal of the radius of the sphere which osculates
in the reduction of the element of the Didonia.
{p.) In general, for any curve on any surface, if £ denote the
vector of the intersection of the axis of the element (or
EXAMPLES TO CHAPTER IX. 193
the axis of the circle osculating to the curve) with the
tangent plane to the surface, then
Hence, for the general Didonia, with the same significa-
tion of the symbols,
£ = p — cTIvdp ;
and the constant c expresses the length of the interval
p— f, intercepted on the tangent plane, between the
point of the curve and the axis of the osculating
circle.
{q.) If, then, a sphere be described, which shall have its centre
on the tangent plane, and shall contain the osculating
circle, the radius of this sphere shall always be equal
to c.
[r.) The recent expression for ^, combined with the first form
of the general differential equation of the Didonia, gives
di = -crdUv Udp ; Vvd^ = 0.
(«.) Hence, or from the geometrical signification of the con-
stant c, the known property may be proved, that if a
developable surface be circumscribed about the arbitrary
surface, so as to touch it along a Didonia, and if this
developable be then unfolded into a plane, the curve
will at the same time be flattened (generally) into a
circular arc, with radius = c.
24. Find the condition that the equation
Sp(<t>+f)-^P=l
may give three real values of y for any given value of p. Ifybe a
function of a scalar, parameter ^, shew how to find the form of this
function in order that we may have
^ ^ dx^ ^ df ^-dz^
Prove that the following is the relation between / and ^,
,.=./• ^f =f^
^ ^{9i+f)i9^+f){9z+f) ^ ^^f
in the notation of § 147.
25. Shew, after Hamilton, that the proof of Dupin's theorem,
that "each naember of one of three series of orthogonal surfaces
cut? each member of each of the other series along its lines of
curvature," may be expressed in quaternion notation as follows :
194 QUATERNIONS.
If Svdp = 0, Sv'dp — 0, S.vv'dp =
be integrable, and if
Svv'= 0, then Fv'dp = 0, makes S.vv'dv = 0.
Or, as follows,
If SvVv = Q, Sv'Vv'=0, Sv"Vv"=:0, and r.w'v"= 0,
then S.v"{Sv'V.v)=:0,
1 „ . d . d J d
where V = i-r-+;T-+«-i--
dx dy dz
26. Shew that the equation
Vap = pVfip
represents the line of intersection of a cylinder and cone, of the
second order, which have /3 as a common generating line.
27. Two spheres are described, with centres at A, B, where
OA = a, OB — y3, and radii a, h. Any line, OFQ,, drawn from the
origin, cuts them in T, Q respectively. Shew that the equation of
the locus of intersection of AT, BQ has the form
r{a + aU{p~a)) {fi + bU(p-fi)) = 0.
Shew that this involves S.a^p = 0,
and therefore that the left side is a scalar multiple of V.afi, so that
the locus is a plane curve.
Also shew that in the particular case
Fal3 = 0,
the locus is the surface formed by the revolution of a Cartesian
oval about its axis.
CHAPTER X.
KINEMATICS.
336.] When a point's vector, p, is a function of the time t, we
have seen (§36) that its vector- velocity is expressed by -j- or, in
Newton's notation, by p.
That is, if p = cpt
be the equation of an orbit, containing (as the reader may see) not
merely the form of the orbit, but the law of its description also, then
p = ^'t
gives at once the form of the Hodograph and the law of its de-
scription.
This shews immediately that the vector-cjcceleration of a point's
motion, d^p
df-"'''
is the vector-velocity in the hodograph. Thus the fundamental pro-
perties of the hodograph are proved almost intuitively.
337.] Changing the independent variable, we have
dp ds ,
P^TsTt^''^'
if we employ the dash, as before, to denote -5- •
This merely shews, in another form, that p expresses the velocity
in magnitude and direction. But a second differentiation gives
p = vp' + v^p".
This shews that the vector-acceleration can be resolved into two
components, the first, vp', being in the direction of motion and
equal in magnitude to the acceleration of the velocity, t; or -=- ;
U/t
the second, v^p", being in the direction of tha radius of absolute
a
196 QUATERNIONS. [338.
curvature, and having for its amount the square of the velocity
multiplied by the curvature.
[It is scarcely conceivable that this important fundamental pro-
position, of which no simple analytical proof seems to have been
obtained by Cartesian methods, can be proved more elegantly than
by the process just given.]
338.] If the motion be in a plane curve, we may write the
equation as follows, so as to introduce the usual polar coordinates,
r and 6, zf
p = ra"^,
where a is a unit-vector perpendicular to, ^ a unit-vector in, the
plane of the curve.
Here, of course, r and may be considered as connected by one
scalar equation ; or better, each may be looked on as a function of i.
By differentiation we get
29 29
p = ra^'^ + rdaa'^ ^,
which shews at once that r is the velocity along, rd that perpen-
dicular to, the radius vector. Again,
2£ 29
which gives, by inspection, the components of acceleration along,
and perpendicular to, the radius vector.
339.] For uniform acceleration in a constant direction, we have at
once, • p = a.
Whence p = ai + l3,
where ^ is the vector-velocity at epoch. This shews that the
hodograph is a straight line described uniformly.
Also p = —-+fit,
no constant being added if the origin be assumed to be the position
of the moving point at epoch.
Since the resolved parts of p, parallel to /3 and a, vary respect-
ively as the first and second powers of i, the curve is evidently a
parabola (§31 (/)).
But we may easily deduce from the equation the following result,
T(p + iPa-^^) =-SUa(p + ^ a-^) ,
the equation of a paraboloid of revolution, whose axis is a. Also
S.a^p = 0,
34I-J xmEMATics. 197
and therefore the distance of any point in the path from the point
— ^/3a~i/3 is equal to its distance from the line whose equation is
Thus we recognise the focus and directrix property.
340.] That the moving point may reach a point y we must
Have, for some real value of t.
Now suppose Ty3, the velocity of projection, to be given =v, and,
for shortness, write ot for Uj3.
Then y = ^i^+viT^.
Since Tzr = 1,
we have («2 _ Say) i^ + Ty'^ = 0,
The values of t'^ are real if
{v^-Say^-Ta^Ty'^
is positive. Now, as TaTy is never less than Say, it is evident that
v^ — Say must always be positive if the roots are possible. Hence,
when they are possible, both values of i^ are positive. Thus we
hscfefoiir values of t which satisfy the conditions, and it is easy to
see that since, disregarding the signs, they are equal two and two,
each pair refer to the same path, but described in ojaposite directions
between the origin and the extremity of y. There are therefore, if
any, in general two parabolas which satisfy the conditions. The
directions of projection are (of course) given by the corresponding
values of ct.
341.] The envelop of all the trajectories possible with a given
velocity, evidently corresponds to
{v^-Sayf-Ta''Ty^ = Q,
for then y is the vector of intersection of two indefinitely close paths
in the same vertical plane.
Now v^ - Say = TaTy
is evidently the equation of a paraboloid of revolution of which the
origin is the focus, the axis parallel to a, and the directrix plane at
a distance ^r- •
la
All the ordinary problems connected with parabolic motion are
easily solved by means of the above formulae. Some, however, are
even more easily treated by assuming a horizontal unit-vector in
198 • QUATERNIONS. [342.
the plane of motion^ and expressing y3 in terms of it and a. But
this must be left to the student.
342.] For acceleration directed to or from a fixed jaoint, we have,
taking that point as origin, and putting P for the magnitude of
the central acceleration,
P =PUp.
Whence, at once, f^pp = 0.
Integrating, Fpp = y = a constant vector.
The interpretation of this simple formula is — first, p and p are in
a plane perpendicular to y, hence the path is in a plane (of course
passing through the origin) ; second, the area of the triangle, two
of whose sides are p and p is constant.
[It is scarcely possible to imagine that a more simple proof than
this can be given of the fundamental facts, that a central orbit is a
plane curve, and that equal areas are described by the radius vector
in equal times.J
343.] When the law of acceleration to or from the origin is that of
the inverse square of the distance, we have
p_ M
Tp"'
where p. is negative if the acceleration be directed to the origin.
Hence p = ^ .
The following beautiful method of integration is due to Hamilton.
(See Chapter IV.)
dJJp Vp.Vpp Up.y
Generally, ^^ = - -^^ =--f^'
, n .. dUp
therefore py = —p. —j- ,
and py = e—pJJp,
where e is a constant vector, perpendicular to y, because
Sy'p = 0.
Hence, in this case, we have for the hodograph,
p = iy"^ — fji,Up.y~\
Of the two parts of this expression, which are both vectors, the
first is constant, and the second is constant in length. Hence the
locus of the extremity of p is a circle in a plane perpendicular to y
(i.e. parallel to the plane of the orbit), whose radius is ^ > and
whose centre is at the extremity of the vector ey""^.
[This equation contains the whole theory of the Circular Hodo-
345-] KINEMATICS. 199
graph. Its consequences are developed at length in Hamilton's
Wem,ents.'\
344. J We may write the equations of this circle in the form
y(p-ey-^) = Yy'
(a sphere), and /Syp =
(a plane through the origin, and through the centre of the sphere).
The equation of the orbit is found by operating by Y.p upon that
of the hodograph. We thus obtain
y = r.pey-i + ^y/Dy-i,
or y2 =Sip + ix.Tp,
or txTp = Se{y^e-'^-p)-,
in which last form we at once recognise the focus and directrix
property. This is in fact the equation of a conicoid of revolution
about its principal axis (e), and the origin is one of the foci. The
orbit is found by combining it with the equation of its plane,
Syp = 0.
We see at once that y^ e^^ is the vector distance of the directrix
. . . Te
from the focus ; and similarly that the eccentricity is — j and the
. -2My^ ''
maior axis — = =- •
345.] To take a simpler case : let the acceleration vary as the dis-
tance from the origin.
Then p = ±m^p,
the upper or lower sign being used according as the acceleration is
from or to the centre.
This is (^ + «.2)p = 0.
Hence p = ae'"«+i3£-™'i
or p = a cos mt + fi sin mt,
where a and j3 are arbitrary, but constant, vectors; and e is the
base of Napier's logarithms.
The first is the equation of a hyperbola (§ 31, ^) of which a and ft
are the directions of the asymptotes ; the second, that of an ellipse
of which a and ft are semi-conjugate diameters.
Since p == m {as'^ — fts'""} ,
or = m {—a sin mt + ft cos mt},
the hodograph is again a hyperbola or ellipse. But in the first
case it is, if we neglect the change of dimensions indicated by the
200 QUATERNIOKS. [346.
scalar factor m, conjugate to the orbit ; in the case of the ellipse it
is similar and similarly situated.
346.] Again, let the acceleration he as the inverse third power of
the distance, we have aUp
Of course, we have, as usual,
Vpp = y.
Also, operating by S.p,
... (xSpp
of which the integral is u
the equation of energy.
Again, Spp = -^ ■
Hence Spp + p'^ = C,
or Spp = Ct,
no constant being added if we reckon the time from the passage
through the apse, where Spp = 0.
We have, therefore, by a second integration,
p^ = Cfi + C' (1)
[To determine C", remark that
pp = Ct + y,
or pV = CH^-y'^.
But p^p^ = Cp^—jj. (by the equation of energy),
= CH^ + CC'-^, by(l).
Hence CC'= ju-y^.]
To complete the solution, we have, by § 133,
where /3 is a unit-vector in the plane of the orbit.
But r^ = - „
p p^
t — _y_.
dt
Hence ^°^~i" ^ "V i
^ ~ '^JCt^ + C
The elimination of t between this equation and (1) gives Tp in
terms of Up, or the required equation of the path.
We may remark that if d be the ordinary polar angle in the
orbit, T/o
log^ = eUy.
348-] KINEMATICS, 201
Hence we have = —Ty I -
W + G' \
and r^=-{Gt''-\-C'\ )
from which the ordinary ec[uations of Cotes' spirals can be at once
found. [See Tait and Steele's Dynamics of a Particle, third edition,
Appendix (A).]
347.] To find the conditions that a given curve may he the hodo-
graph corresponding to a central orbit.
If or be its vector, given as a function of the time, f^ndt is that of
the orbit ; hence the requisite conditions are given by
Tvjftsdt ■=■ y,
where y is a constant vector.
We may transform this into other shapes more resembling the
Cartesian ones.
Thus ^ FijfijTdt = 0,
and * VzrfiiTdt+Vm'ST = 0.
From the first f'^dt = x-ir,
and therefore xYTsis = jf,
or the curve \& plane. And
m T^is + VisTs = ;
or eliminating x, yViim = —(Fm-wY-
Now if v' be the velocity in the hodographj 2if its radius of curva-
ture, p' the perpendicular on the tangent ; this equation gives at
once hv'= R'p"^,
which agrees with known results.
348.] The equation of an epitrochoid or hypotrochoid, referred to
the centre of the fixed circle, is evidently
p = ai " a + M " a,
where a is a unit-vector in the plane of the curve and i another
perpendicular to it. Here o> and co^ are the angular velocities in
the two circles, and t is the time elapsed since the tracing point
and the centres of the two circles were in one straight line.
Hence, for the length of an arc of such a curve,
s —fT'pdt =fdt V { (o^a 2 + 2 tocoi a* cos (o) - (Oi) i! + ffli^ «2 } ^
= I dt ^y \{Q)a + <i)-J)f ±'i:u>a>^ah . ^ — o""^ 3'
which is, of course, an elliptic function.
202 QUATERNIONS. [349-
But when the curve becomes an epicycloid or a hypocycloid,
coa+w^J = 0, and
which can be expressed in finite terms, as was first shewn by Newton
in the Principia.
The hodograph is another curve of the same class, whose equa-
tion is 2iat 2a)j<
p = i{aa)i " a + bco-yi ^ a);
and the acceleration is denoted in magnitude and direction by the
vector iat 2M\t
p = —au?i " a—ba\ i " a.
Of course the equations of the common Cycloid and Trochoid may
be easily deduced from these forms by making a indefinitely great
and o) indefinitely small, but the product aa> finite ; and transferring
the origin to the point p=. aa.
349.] Let i be the normal-vector to any plane.
Let la- and p be the vectors of any two points in a rigid plate in
contact with the plane. •
After any small displacement of the rigid plate in its plane, let
dm and dp be the increments of m and p.
Then Sidm = 0, Sidp = ; and, since T^sr — p) is constant,
S{-a-—p) idsr—dp) = 0.
And we may evidently assume
dp ■= a)i{p — t),
dsT =: Q>J(or — t) ;
where of course t is the vector of some point in the plane, to a rota-
tion 0) about which the displacement is therefore equivalent.
Eliminating it, we have
d('ST — p)
m = -^ —)
•ST — p
which gives to, and thence r is at once found.
For any other point a- in the plane figure
Sida- = 0,
S{p—(t) {dp— da) = 0. Hence dp— da = aji«(p— o-).
S{(T—'m){dijr — d(T) = 0. Hence dcr—dzr = oo^i^a—w).
From which, at once, coj = w^ = co, and
da- = (0? ((T— t),
or this point also is displaced by a rotation a> about an axis through
the extremity of r and parallel to i.
35 1 -J KINEMATICS. 203
•
350.] In the ease of a rigid body moving about a fixed point
let OT, p, a- denote the Ysctors of any three points of the body ; the
fixed point being origin.
Then ■sr', p^, <j^ are constant^ and so are Sssp, Spar, and Sa-sr.
After any small displacement we have, for tn- and p,
Smdzs- = 0, ^
Spdp = 0, i (1)
Szjdp + Spdzj = 0. )
Now these three equations are satisfied by
«?sr =: VazT, dp = V^ap,
where a is any vector whatever. But if dur and dp are given, then
Vdsrdp ^ T^.FazrFap = aS.ap^.
Operate by S.V^p, and remember (1),
S^zrdp = S^pd^ = S^.apTn:
Vdvrdp Vdpdxs ,.
^^^'^^ «= -s^ = -s^ ' ^')
Now consider o-, Strdv = 0, \
Spdcr = —Strdp, V
Sssda = — Sa-duT. )
da = Va<T satisfies them all, by (2), and we have thus the proposi-
tion that ani/ small displacement of a rigid body abont a ^xed point is
equivalent to a rotation.
351.1 To represent the rotation of a rigid body about a given awis^
ihrougli a given finite angle.
Let a be a unit-vector in the direction of the axis, p the vector
of any point in the body with reference to a fixed point in the axis,
and 6 the angle of rotation.
Then p = a''^Sap + a-^Vap,
=■ — aSap — a Vap.
The rotation leaves, of course, the first part unaffected, but the
second evidently becomes
— a ^ aVap,
or — a Vap cos 6 + Vap sin 6.
Hence p becomes
pj z= — aSap — a Vap COS -f- Vap sin d,
= (cos- + asm-jp(cos--asm-j.
= a pa
204 QUATERNIONS. [352.
352.] Hence to compound two rotations about axes which meet, we
may evidently write, as the effect of an additional rotation <^ about
the unit-vector ;8, ^ _*
Hence p^ = P' a" pa~ " p~' .
If the /3-rotation had been first, and then the a-rotation, we should
have had 1 ± _* _i.
and the non-commutative property of quaternion multiplication
shews that we have not, in general,
P'i = ft-
If a, fi, y be radii of the unit sphere to the corners of a spherical
triangle whose angles are - > ^ . - > we know that
U &i u ^
y" /3 " o "^ = — 1 . (Hamilton, Lectures, p. 267.)
if. *
Hence /3'o'=— y"'',
-t ±
and we may write P2 = y " py^^
or, successive rotations about radii to two corners of a spherical triangle,
-and through angles double of those of the triangle, are equivalent to a.
single rotation about the radius to the third corner, and through an
angle double of the exterior angle of the triangle.
Thus any number of successive _/?«Jfe rotations may be compounded
into a single rotation about a definite axis.
353.] When the rotations are indefinitely small, the effect of
one is, by § 351, p^ = f,-\-OiVap,
and for the two, neglecting products of small quantities,
p^ = p-\-(xTap+W^P,
a and b representing the angles of rotation about the unit-vectors
a and ^ respectively.
But this is equivalent to
P2 = p + r(aa-hb^)FV(aa + bi3)p,
representing a rotation through an angle T{fi,a + b^), about the unit-
vector TJ((xa + 6)3). Now the latter is the direction, and the former
the length, of the diagonal of the parallelogram whose sides are
(xa. and b/8.
We may write these results more simply, by putting a for <ya,
/3 for b/3, where a and ^ are now no longer unit-vectors, but repre-
355-1 KINEMATICS. 205
sent by their versors tlie axes, and by their tensors the angles (small)j
of rotation.
Thus pj^ = p+ Vap,
P2 = p+Fap+ V^p,
= p+Fia + p)p.
354.] The general theorem, of which a few preceding sections
illustrate special cases, is this :
By a rotation, about the axis of q, through double the angle of q,
the quaternion r becomes the quaternion qrq~^. Its tensor and
angle remain unchanged, its plane or axis alone varies.
A glance at the figure is sufficient for . q
the proof, if we note that of course
T.qrq"^^ Tr, and therefore that we need
consider the versor parts only. Let Q
be the pole of q,
A£=q, JJB' = q-\ WC' = r.
Join C'A, and make AG = C'A. Join
CB.
Then CB is qrq-'^, its arc CB is evidently equal in length to that
of r, B'C; and its plane (making the same angle with B'B that
that of B'C does) has evidently been made to revolve about Q, the
pole of q, through double the angle of q.
If r be a vector, = p, then qpq"^ (which is also a vector) is the
result of a rotation through double the angle of q about the axis
of q. Hence, as Hamilton has expressed it, if B represent a rigid
system, or assemblage of vectors,
qBq-<^
is its new position after rotating through double the angle of q
about the axis of q.
355.] To compound such rotations, we have
r.qBq'^.r^''- =rq.B.{rq)-^.
To cause rotation through an angle ^-fold the double of the angle
of q we write q^Bq-K
To reverse the direction of this rotation write q~^BqK
To translate the body B without rotation, each point of it moving
through the vector a, we write a + B.
To produce rotation of the translated body about the same axis,
and through the same angle, as before,
q{a + B)q-\
Had we rotated first, and then translated, we should have had
a + qBq-'^.
206
QUATERNIONS.
[356.
The obvious discrepance between these last results might perhaps
be useful to those who do not believe in the Moon's rotation, but
to such men quaternions are unintelligible.
356.] Given the instantaneous axis in terms of the time, it is re-
quired to find the single rotation which will bring the body from any
initial position to its position at a given time.
If a be the initial vector of a point of the body, ot the value of
the same at time t, and q the required quaternion, we have
^ = i°r^ (1)
Differentiating with respect to t, this gives
■ar = qaq~^—qaq''^qq~^,
= 2r.{rqq-\qaq-^).
But ■a^ = Vei!7 = V.eqaq~^.
Hence, as qaq"^ may be any vector whatever in the displaced
body, we must have e = 2 Tqq-^ (2)
This result may be stated in even a simpler form than (2), for we
have always, whatever quaternion q may be,
dUq
Vqq-^ =
dt
{Vq)-
and, therefore, if we suppose the tensor of q, which may have any
value whatever, to be a constant (unity, for instance), we may write
(2) in the form eq = 2q (3)
An immediate consequence, which will be of use to us later, is
q.q-''eq = 2q (4)
357.J To express q in terms of the usual angles i/f, 6, ^.
Here the vectors i, J, h in the original position of the body corre-
•spond to OA, OB, 00, respectively,
at time t. The transposition is ef-
fected by — first, a rotation y]r about
k ; second, a rotation 6 about the
new position of the line originally
coinciding with/; third, a rotation
(^ about the final position of the line
at first coinciding with k.
Let i, j, k be taken as the initial
directions of the three vectors which
at time t terminate at A, B, C re-
spectively.
The rotation >/f about h has the operator
t _i
k''{ )k ''.
357-] KINEMATICS. 207
This converts y into r), where
t -i- .
'tj = k''ji " = J COS yp—i sin \{r.
The body next rotates about tj through an angle 9. This has
the operator t _*
It converts k into
^ * _^ 6 Q Q 6
OC = C= yfk-q "= (cos- +'?sin-)/4(cos-— jjsin-)
= ^cos04sind(jcos\/f + ysin\/f).
The body now turns through the angle (p about C the operator
being * _*
Hence
= (eos - + C sin -) (cos - + j? sin -) (cos| + k sin |)
= [cos~ + Csm-) cos-cos^ + Acos-sin|^
. -Jf , . . . , . 6 . if ,. . . ~[
+ sin-cos^(_;cosT//— »sini|f) + sin-sin^(«eosi/f+_;smi/f)
a 2 2 2 _J
/ <^..^sr e ^// . . e . f . . e ^ir , ■ ■^i
= (cos-+Csin-) cos-cos^ — »sm-sin-i^ + » sin -cos- + /ecos-sin—
\ 2 ^ 2''L2 2 2 2-' 2 2 2 2j
4> ^ . (b . 6 . ■Jf . „
= cos — cos -cos— + sm — sin-sin— sm^cosiir
222222 ^
. A . e ^ ■ „ . , .</> e . ■jf
— sm— sm-eos — sin5sm\/f— sm — cos- sin — cos ^
2 2 2 ^ 2 2 2
.^ (j) . 6 . -f . <j) 6 ^ . „
+ «( — cos — sin - sin — + sm — cos - cos — sm 6 cos <|f
V 2 222 2'2 ^
. d) . 6 -Jf „ . (b e . \1/ . „ . N
— sm — sm - cos ^ cos 9 + sm — cos - sin — sm sin i/j )
• 2 2 2 2 2 2 ^■'
.f (b . d lif . (b e ^ . „ .
+ f I cos — sm - eos — + sm — cos - eos — sm a sm \|f
■^^222222 ^
. <b . e . f „ . <f) . -^ . „ ,\
— sm — sin - sm j- cos 0— sin — cos - sin sm 6 eos i/f )
2 2 2 2 2 2 ^>'
7/0 . yj/ . <l) if a
+ ^ I cos — cos - sin — + sm ^ cos - cos — cos 9
V2 2 '2 2 2 2
.rf).e.>|f.^., . (h . ir . „ x
+ sm5-sin-sin — sm0sinvf+ sm— sin- cos— sm9cos\/r)
222 ^222 ^/
rf) + i/f 5 . . (b—yif . . (b—yJf . , . <b + ^lt 6
= cos cos - + « sm sm - +_; cos sin- + a sm -^^ — 2: cos- >
which is, of course, essentially unsyuimetrical.
208 QUATERNIONS. [358.
358.] To find the usual equations connecting \j/, 6, (p with the an-
gular velocities about three rectangular axes fixed in the body.
Having tlie value of q in last section in terms of the three angles,
it may be useful to employ it, in conjunction with equation (3) of
§ 356, partly as a verification of that equation. Of course, this is
an exceedingly roundabout process, and does not in the least re-
semble the simple one which is immediately suggested by qua-
ternions.
We have 2q=. eq= {<a^OJ + oo20£ + <agOC} q,
■whence ^i~^4 = S~^ {o)iOA + ai^ OB + co^ 00} q,
or 2q = q{ia)i +ju>2 + ka^).
This breaks up into the four (equivalent to three independent)
equations
2 -7; ( cos ^^-—-!- cos-)
dt^ 2 2>
. d)—-J/ . 6 d> — -Jf . e . (b + \lf e
= — CO, sm — — -^ sin - — o), cos - — — sin co„ sm - — — cos - j
^2 2^2 2^2 2
2-T:(sin ^„ ^ sin-)
dt V 2 2''
(t> + \l/ e . (b + xir 6 (h — ylr .
= 0), cos — - — cos- — coosm — — ^eos- + a),cos-^^ — i-sm-,
1 2 2 ^ 2 2 ^ 2 2
^d ^ (b — \lr . 0\
2 — (cos „ sin-l
dt\ 2 2'
. (t> + ^ (b + ylf . tb — yS, .
= Q)j sin cos - + i»2 cos cos ;r — 0)3 Sin — — ~ sm - J
2 ^ ( sin ^ cos - )
dt^ 2 2'
(b—-d/ . . <b — ^ . (h + y],
= —6)1 cos sin- + 6)2 sin — —^ sm- + a>3C0S^!^ — 31 cos--
Prom the second and third eliminate 0— x/^, and we get by in-
spection ^ a / ■ i , ,^ ^
^ COS - . = (uj sm <p + 0)2 cos (^) cos - >
or ^ = Wj sin (/) + Wg cos (/). (1)
Similarly, by eliminating between the same two equations,
. 0,- -.s ■ ^ ^ ^ • ^
sm — ((^ — \\r) = cog sin — + iBi cos 9 cos — — a>^ sm <p cos— •
2 i £1 2
And from the first and last of the group of four
■ ■ . . .
cos-{(l> + \j/) = 0)3 cos- — WjCosc^ sm- + Wg sm <^ sm - •
2 2 i ^
359-] KINEMATICS. 209
These last two equations give
<j) + \jfCOsd = 0)3 (2)
<p cos 6 + \jf = ( — w^ cos (^ + 0)2 sin (^) sin + 0)3 cos0.
From the last two we have
■\jr sin 6 =— ctfj 008(^ + 0)2 sin (^ (3)
(1), (2), (3) are the forms in which the equations are usually given.
359.] To deduce eiepressions for the direciion-coaines of a set 0/
rectangular axes in any position in terms of rational functions of three
Let a, yS, y be unit- vectors in the directions of these axes. Let c[
be, as in § 356, the requisite quaternion operator for turning the
coordinate axes into the position of this rectangular system. Then
q^ = w + xi-^yj-^zh,
where, as in § 356, we may write
1 = W^+iB^+y^+^2.
Then we have (f^ =■ w—xi—yj-\-zk,
and therefore
a = qiq"^ = {wi—x—yk + zf){w—xi—yj—z}c)
= [ie^ +x^ —y'^ —z^)i+ 2 {wz + xy)J +2(xz—toy)k,
where the coefficients of i, J, k are the direction-eosines of a as
required. A similar process gives by inspection those of ^ and y.
As given by Cayley*, after Rodrigues, they have a slightly
different and somewhat less simple form — to which, however, they
are easily reduced by putting
_'^_.5'_'^_ ^
\ jJ. V ^i
The geometrical interpretation of either set is obvious from the
nature of quaternions. For (taking Cayley's notation) if be the
angle of rotation : cos^ cosy, cos h, the direction-cosines of the axis,
we have
6 6
q = w + xi+yj + zJc = cos- + sin- (i cos/ +/ cosy + /i cos ^),
Q
SO that w — cos - >
X = sm-cos/,
a
.
y = sm-cosy;
• ^ i
z = sm - cos n.
2
* Camb. and Bub. Math. Journal. Vol. i. (1846.)
210 QUATERNIONS. [360.
From these we pass at once to Rodrigues' subsidiary formulae,
K = -5 = sec^ - .
w^ 2
X = — = tan - cos/,
&c. = &c.
360.J By the definition of Homogeneous Strain, it is evident that
if we take any three (non-eoplanar) unit-vectors a, /3, y in an un-
strained mass, they become after the strain other vectors, not neces-
sarily unit- vectors, a^, ySj^, y^.
Hence any other given vector, which of course may be thus ex-
pressed, p =i xa + yfi -\- zy,
becomes Pi = c(ia-^-^y^^->r zy-^,
and is therefore known if a^, j3^, yj be given.
More precisely
pS.afiy = aS.j3yp + j3S.yap + yS.al3p
becomes
piS.a^y = (ppS.a^y = aj^S.^yp + ^-j^S.yap + yiS.a^p.
Thus the properties of cf), as in Chapter V, enable us to study with
great simplicity strains or displacements in a solid or liquid.
For instance^ to find a vector whose direction is unchanged hy the
strain, is to solve the equation
Yp^p = 0, or <^p = gp,
where ^ is a scalar unknown.
[This vector equation is equivalent to three simple equations, and
contains only three unknown quantities ; viz. two for the direction
of p (the tensor does not enter, or, rather, is a factor of each side),
and the unknown ^.]
We have seen that every such equation leads to a cubic in g
which may be written
g^—m^g'^ + m^g—m = 0,
where ««2i ^u ''"■ ^'"^ scalars depending in a known manner on the
constant vectors involved in 0. This must have one real root, and
may have three.
361. J For simplicity let us assume that a, /3, y form a rectangular
system, then we may operate by S.a, S.^, and S.y; and thus at
once obtain the equation for g, in the form
0... (1)
Saoj^ 4 g.
Sa^i,
Say^
S^a^,
Sl3^i+ff,
S^y,
Syai,
SyPi,
^yyi+9
362.J
KINEMATICS.
211
To reduce this we have
'S'aoi, Sa^i, Say^^
S^a^, Sj3p^, Sl3y^
Sya^, Sy^i, Syy^
1 S^aa^ + S^pa^ + S^ya,, ^Saa^SajS^,
Sya^, Syl3i,
which, if the mass be rigid, becomes successively
^Saa^Sayi
Sl3y,
Syy^
Saa^ \Sypj^,
mi
= s^mMyri-7iSyPi)
Thus the equation becomes
- 1 -ff{Saa^ + aS^^i + Syy^) +g^ {Saa^ + Sp^^ + Syy^) +g^ = 0,
{g-^){9^+9{^+Saa^ + Spp^ + 8yy;)+l) =0.
362.] If we take Tp :=G we consider a portion of the mass
initially spherical. This becomes of course
or
n-
C,
an ellipsoid, in the strained state of the body.
Or if we consider a portion which is spherical after the strain, i. e
Tp^ = C,
its initial form was T^p = C,
another ellipsoid. The relation between these ellipsoids is obvious
from their equations. (See § 311.)
In either case the axes of the ellipsoid correspond to a rectangular
set of three diameters of the sphere (§ 257). But we must care-
fully separate the cases in which these corresponding lines in the
two surfaces are, and are not, coincident. For, in the former case
there is jmre strain, in the latter the strain is accompanied by ro-
tation. Here w6 have at once the distinction pointed out by
Stokes* and Helmholtzf between the cases of fluid motion in
which there is, or is not, a velocity-potential. In ordinary fluid
motion the distortion is of the nature of a pure strain, i.e. is differ-
entially non-rotational ; while in vortex motion it is essentially ac-
companied by rotation. But the resultant of two pure strains is
generally a strain accompanied by rotation. The question before us
beautifully illustrates the properties of the linear and vector function.
* Cambridge Phil Trans. 1845.
+ Crelle, vol. Iv. 1857. See also Phil Mag. (Supplement) June 1867.
P 2
212 QUATEENIONS. [363.
363.] To find the criterion of a pure strain. Take a, p, y now as
unit-vectors parallel to the axes of the strain-ellipsoid, they become
after the strain a a, bj3, cy.
Hence p, = (pp ——aaSap—b^S^p — cySyp.
And we have, for the criterion of a pure strain, the property of the
function <\>, that it is self-conjugate, i. e.
Sp(fi<T = S(T<pp.
364.J Two pure strains, in succession, generally give a strain ac-
companied hy rotation. For if <p, \jf represent the strains, since they
are pure we have Sp^a- = Sai^tp, ^
But for the compound strain we have
Pi = XP = ^^P>
and we have not generally
Spx<T = Saxp.
For 8p^<ji<T = Sa-(j)\jfp,
by (1), and i/?0 is not generally the same as (f)\j/. (See Ex. 7 to
Chapter V.)
365.] The simplicity of this view of the question leads us to
suppose that we may easily separate the purs strain from the rotation
in any case, and exhibit the corresponding functions.
When the linear and vector function expressing a strain is self-
conjugate the strain is pure. When not self-conjugate, it may be
broken up into pure and rotational parts in various ways (analogous
to the separation of a quaternion into the sum of a scalar and a
vector part, or into the product of a tensor and a versor part), of
which two are particularly noticeable. Denoting by a bar a self-
conjugate function, we have thus either
€l> = qw{ )q-\ or 4, = ^^.q{ )q-\
where e is a vector, and q a quaternion (which 'may obviously be
regarded as a mere versor).
That this is possible is seen from the fact that (j) invofves nine
independent constants, while ^ and ct each involve six, and e and q
each three. If </>' be the function conjugate to <f>, we have
<j>'=f-F.e{ ),
60 that 2\}r = <p + <f)',
and 2r.i{ ) = 0-()i',
which completely determine the first decomposition. This is, of
365.] KINEMATICS. 213
course, perfectly well known in quaternions, but it does not seem
to have been noticed as a theorem in the kinematics of strains that
there is always one, and but one, mode of resolving a strain into the
geometrical composition of the separate effects of (1) a pure strain,
and (2) a rotation accompanied by uniform dilatation perpendicular
to its axis, the dilatation being measured by (sec. 0—\) where Q is
the angle of rotation.
In the second form (whose solution does not appear to have been
attempted), we have
<t> = i^{ ) r\
where the pure strain precedes the rotation, and from this
0'=:^.j-i( )q,
or in the conjugate strain the rotation (reversed) is followed by the
pure strain. From these
and OT is to be found by the solution of a biquadratic equation*.
It is evident, indeed, from the identical equation
S.CT<p'(l)p = S.p(j/(I><T
that the operator ^'^ is self-conjugate.
In the same way
<^<^'( )=q^^{s-H )q)q-\
or §■-! {4,<t)'p) q = ^^ iq-'^pq) = ¥^ [q'^pq),
which shew the relations between ^<^', <^'0, and q.
To determine q we have
<t>p-q = q^P
* Suppose the cubic in ct to be
ra^' + 3^"" + gr, TO^ + 32 = 0,
write 0; for ^'<j> in the given equation, and by its help this may be written as
(W + sf)a) + jriW= + g'2 = = w'(o) + g',)+araj + 5f2.
Eliminating 5=, we have
<"' + (2?, -ff") oi' + {g,''-2gg,)o,-g^ = 0.
This must agree with the (known) cubic in ai,
0^ -i- mar' + m^a + ma=0,
suppose, so that by comparison of coefficients we have
so that g, is known, and g= ' •
2 -v/— ma
where 2^. = m-(^^^
The values of the quantities g being found, w is given in terms of <u by the equation
above. (Proc. B. S. E., 1870-71-)
214 QUATERNIONS. [366.
whatever be p, so that
S.Fq{<f)—m)p= 0,
or S.p{<^'-^)Fq = Q,
which gives {'¥~ ^) ^i = 0>
The former equation gives evidently
whatever be o and /3 ; and the rest of the solution follows at once.
A similar process gives us the solution when the rotation precedes
the pure strain.
366.] In general, if
Pi = (jyp = —CiSap—^j^S^p—y-^Syp,
the angle between any two lines, say p and a; becomes in the
altered state of the body
cos-^ {-S.U<l)pU<l><T).
The plane *Sfp = becomes (witji the notation of § 144)
SCpi = = SC<l>p = Sp<l>'C
Hence the angle between the planes SCp = 0, and Srjp = 0, which
is cos~^(—iS.UCUri), becomes
cos-^{-S.U<l>'CU<l,'ri).
The locus of lines equally elongated is, of course,
T^Up = e,
or T<i,p = eTp,
a cone of the second order.
367.] In the case of a Simple Shear, we have, obviously,
Pi = <i>P = p + fiSap,
where Sa^ =0.
The vectors which are unaltered in length are given by
Tp^ = Tp,
or 2 S^pSap + l3^S^ap = 0,
which breaks up into S. ap = 0,
and Sp{2fi + fi^a) = 0.
The intersection of this plane with the plane of a, /3 is perpen-
dicular to 2/3 + /3*a. Let it be a + a? /3, then
-S.(2/3 + y32a)(a + a;/3) = 0,
i.e. 2a! — 1 = 0.
Hence the intersection required is
368.] KINEMATICS. 215
<^
For the axes of the strain, one is of course aj3, and the others
are found by making TcjyJJp a maximum and minimum.
Let p = a + x^,
then pi= (pp = a + xj3—l3,
and -^ = max. or mm.,
Tp
1
gives x^—x+-^ = 0,
from which the values of x are found.
Also, as a verification,
S.{a + XiP){a + X2l3) =—l + p.'^x^x^,
and should be = 0. It is so, since, by the equation,
_ 1
Again
S{a + {x^-l)fi} {a-\-{x^-\)p} =-\+&^{x^x^-{x^ + x.,)+l},
which ought also to be zero. And, in fact, aj^ + ^g = 1 by the equa-
tion ; so that this also is verified.
368.] We regret that our limits do not allow us to enter farther
upon this very beautiful application.
But it may be interesting here, especially for the consideration
of any continuous displacements of the particles of a mass, to in-
troduce another of the extraordinary instruments of analysis which
Hamilton has invented. Part of what is now to be given has been
anticipated in last Chapter, but for continuity we commence afresh.
If Fp = C (1)
be the equation of one of a system of surfaces, and if the differential
of (l)be Svdp= 0, (2)
v is a vector perpendicular to the surface, and its length is inversely
proportional to the normal distance hetween two consecutive surfaces.
In fact (2) shews that v is perpendicular to dp, which is any tangent
vector, thus proving the first assertion. Also, since in passing to a
proximate surface we may write
Svbp = 8C,
we see that F{p + v-^hC) = C + W.
This proves the latter assertion.
It is evident from the above that if (1) be an equipotential, or an
isothermal, surface, —v represents in direction and magnitude the force
at any point or the flux of heat. And we have seen (§ 317) that if
. d . d -. d
dx '' dy dz
216 QUATERNIONS. [369.
d'' A^ d^
gmng v^=______,
then V = VFp.
This is due to Hamilton (Lectures on Quaternions, p. 611).
369.] From this it follows that the effect of the vector operation
V, upon any scalar function of the vector of a point, is to produce
the vector which represents in magnitude and direction the most rapid
change in the value of the function.
Let us next consider the effect of V upon a vector function as
<^ = ii+Jv + ^C-
We have at once
-=-{g-$-S)-'(S-f)-^-
and in this" semi-Cartesian form it is easy to see that : —
If T represent a small vector displacement of a point situated at
the extremity of the vector p (drawn from the origin)
SV a- represents the consequent cubical compression of the group
of points in the vicinity of that considered, and
VVa represents twice the vector axis of rotation of the same
group of points.
Similarly 5. V= - (^^ +, i- + C^) = -D.,
or is equivalent to total differentiation in virtue of our having
passed from one end to the other of the vector a.
370.] Suppose we fix our attention upon a group of points which,
originally filled a small sphere about the extremity of p as centre,
whose equation referred to that point is
To3 = e (1)
After displacement p becomes p + a-, and, by last section, p + a>
becomes p + m + cr— (jSa)V)o-. Hence the vector of the new surface
which encloses the group of points (drawn from the extremity of
p + tr) is Q)i = oi — {8<i>V)(T (2)
Hence o) is a homogeneous linear and vector function of w-^ ; or
and therefore, ^7 (1)> ^^o)i = e,
the equation of the new surface, which is evidently a central surface
of the second order, and therefore, of course, an ellipsoid.
We may solve (2) vsdth great ease by approximation, if we re-
member that T^ is very small, and therefore that in the small term
we may put <Bj for w ; i. e. omit squares of small quantities ; thus
(o = <Bj + (Sa>jV)a:
372-] KINEMATICS. '217
371.] If the small- displacement of each point of a medium is in the
direction of, and proportional to, the attraction exerted at that point
hy any system of material masses, the displacement is effected without
rotation.
For \i Fp = C be the potential surface^ we have Sddp a complete
differentia] ; i. e. in Cartesian coordinates
^dx + r]di/ + (dz
is a differential of three independent variables. Hence the vector
axis of rotation ^ ^^ g
vanishes by the vanishing of each of its constituents, or
r.Va- = 0.
Conversely, if there he no rotation, the displacements are in the
direction of and proportional to, the normal vectors to a series of
surfaces.
For 0=r.dpr.Vcr = (SdpV) a- - ^Sadp,
where, in the last term, V acts on o- alone.
Now, of the two terms on the right, the first is a complete differ-
ential, since it may be written —Dcip(T, and therefore the remaining
term must be so.
Thus, in a distorted system, there is no compression if
SVa- = 0,
and no rotation if V.Va = ;
and evidently merely transference if o- = a = a constant vector,
which is one case of Vg- = q.
In the important case of a- = eVFp
there is evidently no rotation^ since .
Vff = eV^Fp
is evidently a scalar. In this case, then, there are only translation
and compression, and the latter is at each point proportional to the
density of a distribution of matter, which would give the potential
Fp. For if r be such density, we have at once
V^Fp = 4 7rr*.
372.] The Moment of Inertia of a body about a unit vector a as
axis is evidently jfp = -■2m{rapf,
where p is the vector of the portion m of the mass, and the origin
of p is in the axis.
« Proc. B. 8. K, 1862-3,
218 QUATERNIONS. [372.
Hence if we take hTa = e^, we have, as locus of the extremity of a,
Jfe* =—^m,{Japf = MSai^a (suppose),
the momental ellipsoid.
If ts be the vector of the centre of inertia, o- the vector of m with
respect to it, we have p = ot + o- ;
therefore MB =-^m{{ Va^f + ( Faaf }
= -M{ Vamf + MSa<i,^a.
Now, for principal axes, Jc is max., min., or max.-min., with the
condition ^z = _ 1 .
Thus we have Sa{'arFaz7—(f)ia) — 0,
Saa = ;
therefore — ^la + wFatiT = ^a = h^o. (by operating by So).
Hence (<^-^-\-k'^-\-vs^)a = +cr<S'aOT (1)
detei-mines the values of a, Ic^ being found from the equation
<St!r(<^ + P + OT2)-lt!7 = 1 (2)
Now the normal to AS(r(0 + P + OT2)-^(7 = 1, (3)
at the point o- is ((/> + ^^ + ot^)"^ o-.
But (3) passes through — sr, by (2), and there the normal is
which, by (1), is parallel to one of the required values of a. Thus
we prove Binet's theorem that the' principal axes at any point are
normals to the three surfaces, eonfocal with the momental ellipsoid,
which pass through that point.
EXAMPLES TO CHAPTER X.
1. Form, from kinematical principles, the equation of the cycloid ;
and employ it to prove the well-known elementary properties of the
arc, tangent, radius of curvature, and evolute, of the curve.
2. Interpret, kinematically, the equation
p = aU{pt-p),
where /3 is a given vector, and a a given scalar.
Shew that it represents a plane curve ; and give it in an in-
tegrated form independent of t.
EXAMPLES TO CHAPTER X. 219
3. If we write ct = ^i—p,
the equation in (2) becomes
/3 — ■nr = aUv7.
Interpret this kinematically ; and find an integal.
What is the nature of the step we have taken in transforming
from the equation of (2) to that of the present question ?
4. The motion of a point in a plane being given, refer it to
{a.) Fixed rectangular vectors in the plane.
{b.) Rectangular vectors in the plane, revolving uniformly
about a fixed point.
(c.) Vectors, in the plane, revolving with different, but uni-
form, angular velocities.
{d.) The vector radius of a fixed circle, drawn to the point of
contact of a tangent from the moving point.
In each case translate the result into Cartesian coordinates.
5. Any point of a line of given length, whose extremities move
in fixed lines in a given plane, describes an ellipse.
Shew how to find the centre, and axes, of this ellipse j and
the angular velocity about the centre of the ellipse of the tracing
point when the describing line rotates uniformly.
Transform this construction so as to shew that the ellipse is a
hypotrochoid.
6. A point. A, moves uniformly round one circular section of
a cone; find the angular velocity of the point, a, in which the
generating line passing through A meets a subcontrary section
about the centre of that section.
7. Solve, generally, the problem of finding the path by which a
point will pass in the least time from one given point to another,
the velocity at the point of space whose vector is p being expressed
by the given scalar function y^.
Take also the following particular cases : —
(a.) fp=.a while Sap> 1,
fp = h while Sap < 1 .
{h.) fp = Sap.
(c.) fp = -p^. (Tait, Trans. R. S. E., 1865.)
8. If, in the preceding question,//) be such a function of Tp that
any one swiftest path is a circle, every other such path is a circle,
and all paths diverging from one point converge accurately in
another. (Maxwell, Gam. and Bub. Math. Journal, IX. p. 9.)
220 QUATERNIONS
9. Interpret, as results of the composition of successive conical
rotations, the apparent truisms
y fi a
and "^i -1^=1.
Kid y p o.
(Hamilton, Lectures, p. 334.)
1 0. Interpret, in the same way, the quaternion operators
} = (8s-')*(<f-')*(f»"')'.
1 1 . rind the axis and angle of rotation by which one given rect-
angular set of unit-vectors a, fi, y is changed into another given
set Oi, Pi, yj.
12. Shew that, if <f>p = p+ Vep,
the linear and vector operation (^ denotes rotation about the vector e,
together with uniform expansion in all directions perpendicular
to it.
Prove this also by forming the operator which produces the
expansion without the rotation, and that producing the rotation
without the expansion ; and finding their joint effect.
13. Express by quaternions the motion of a side of one right
cone rolling uniformly upon another which is fixed, the vertices of
the two being coincident.
14. Given the simultaneous angular velocities of a body about
the principal axes through its centre of inertia, find the position
of these axes in space at any assigned instant.
15. Find the linear and vector function, and also the quaternion
operator, by which we may pass, in any simple crystal of the
cubical system, from the normal to one given face to that to an-
other. How can we use them to distinguish a series of faces be-
longing to the same zone ?
16. Classify the simple forms of the cubical system by the
properties of the linear and vector function, or of the quaternion
operator.
17. Find the vector normal of a face which truncates symmetri-
cally the edge formed by the intersection of two given faces.
18. Find the normals of a pair of faces symmetrically truncating
the g^ven edge.
EXAMPLES TO CHAPTER X. 221
19. Find the normal of a lace which is equally inclined to three
given faces.
20. Shew that the rhombic dodecahedron may be derived from
the cube, or from the octahedron, by truncation of the edges.
2 1 . Find the form whose faces replace, symmetrically^ the edges
of the rhombic dodecahedron. ♦
22. Shew how the two kinds of hemihedral forms are indicated
by the quaternion expressions.
23. Shew that the cube may be produced by truncating the edges
of the regular tetrahedron.
24. Point out the modifications in the auxiliary vector function
required in passing to the pyramidal and prismatic systems re-
spectively.
25. In the rhombohedral system the auxiliary quaternion operator
assumes a singularly simple form. Give this form, and point out
the results indicated by it.
26. Shew that if the hodograph be a circle, and the acceleration
be directed to a fixed point ; the orbit must be a conic section,
which is limited to being a circle if the acceleration follow any other
law than that of gravity.
27. In the hodograph corresponding to accelerationy(Z') directed
towards a fixed centre, the curvature is inversely as D^y^D).
28. If two circular hodographs, having a common chord, which
passes through, or tends towards, a common centre of force, be cut
by any two common orthogonals, the sum of the two times of hodo-
graphically describing the two intercepted arcs (small or large) will
be the same for the two hodographs. (Hamilton, Mements, p. 725.)
29. Employ the last theorem to prove, after Lambert, that the
time of describing any arc of an elliptic orbit may be expressed in
terms of the chord of the arc and the extreme radii vectores.
30. If $'( )s~^ be the operator which turns one set of rect-
angular unit- vectors a, /3, y into another set oj, /3^, y^, shew that
there are three equations of the form
CHAPTER XI.
PHYSICAL APPLICATIONS.
373.] We propose to conclude the work by giving a few in-
stances of the ready appHcability of quaternions to questions of
mathematical physics, upon which, even more than on the Geo-
metrical or Kinematical applications, the real usefulness of the
Calculus must mainly depend — except, of course, in the eyes of that
section of mathematicians for whom Transversals and Anharmonic
Pencils, &c. have a to us incomprehensible charm. Of course we
cannot attempt to give examples in all branches of physics, nor
even to carry very far our investigations in any one branch : this
Chapter is not intended to teach Physics, but merely to shew by
a few examples how expressly and naturally quaternions seem to be
fitted for attacking the problems it presents.
We commence with a few general theorems in Dynamics — the
formation of the equations of equilibrium and motion of a rigid
system, some properties of the central axis, and the motion of a solid
about its centre of inertia.
374.J When any forces act on a rigid body, the force /3 at the
point whose vector is a, &c., then, if the body be slightly displaced,
so that a becomes a + 6 a, the whole work done is
28pba.
This must vanish if the forces are such as to maintain -equilibrium.
Henoe ike condition of equilibrium of a rigid body is
2 SjSha = 0.
For a displacement of translation 8a is ani/ constant vector, hence
2/3 = (1)
For a rotation-displacement, we have by § 350, e being the axis,
and Ti being indefinitely small,
6a = Ft a.
377-J PHYSICAL APPLICATIONS. 223
and S/S.^Tea = S/S.fTa/S = S.eliFafi) = 0,
whatever be e, hence 2 . Ta^ = (2)
These equations, (1) and (2), are equivalent to the ordinary six
equations of equihbrium.
375.] In general, for any set of forces, let
2/3 = /3i,
2.ra/3 = ai,
it is required to find the points for which the couple a-^ has its axis
coincident with the resultant force ^^. Let y be the vector of such a
point.
Then for it the axis of the couple is
2.F(a-y)^ = ai-ry^i,
and by condition x^-^ = a^ — Fy/Sj .
Operate by S^-^ ; therefore
x^l ^ ^ai/3i,
and Ty^i = a^ -ft-^^iA = -^Ja^^^-^,
or y = ^«i/3r^+.5'i3i,
a straight line (the Central Axis) parallel to the resultant force.
376.] To find the points about which the couple is least.
Here T{a^— Vyl3j) = minimum.
Therefore S. (a^— FyjSj) F^iy = 0,
where y' is any vector whatever. It is useless to try y'= ^^, but
we may put it in succession equal to a^ and Vai^^. Thus
S.yr.0^ra^P^ = Q,
and {ra^^yf-fi\S.yra^p^ = 0.
Hence y = x Va^ /Sj + j^/Sj ,
and by operating with S.Va^^^, we get
Pi
or y= ra^Py-"^ +y/3i,
the same locus as in last section.
377.] The couple vanishes if
«i- ^7/8i = 0.
This necessitates Sa^fi^ = 0,
or the force must be in the plane of the couple. If this be the case,
still the central axis.
224 QUATERNIONS. [378.
378.] To assign the values of forces £, i^, to act at «, ej, and be
equivalent to the given system.
Hence Fe^H- n^ifi^-i) = a^,
and i = (e- ei)-i (a^ - Tei ^1) + a; (e - €1).
Similarly for f^. The indefinite terms may be omitted, as they
must evidently be equal and opposite. In fact they are any equal
and opposite forces whatever acting in the line joining the given
points.
. 379.] For the motion of a rigid system, we have of course
^S{md—/3)ba = 0,
by the general equation of Lagrange.
Suppose the displacements 6a to correspond to a mere translation,
then 8a is an^ constant vector, hence
'2{md — 0) = 0,
or, if ai be the vector of the centre of inertia, and therefore
a^'Em = 'Ema,
we have at once di'Sm — 2/3 = 0,
and the centre of inertia moves as if the whole masa were concen-
trated in it, and acted upon by all the applied forces.
380.] Again, let the displacements 8 a correspond to a rotation
about an axis «, passing through the origin, then
ba = Fea,
it being assumed that Te is indefinitely small.
Hence I,S.eFa{m'd—^) = 0,
for all values of e, and therefore
I,.Fa{md-0) = 0,
which contains the three remaining ordinary equations of motion.
Transfer the origin to the centre of inertia, i. e. put a = a^ + ot,
then our equation becomes
2r(a, + in-) (jKiii + »««■— /3) = 0.
Or, since 2»»ot = 0,
2 Fot (»» OT - y3) + Fai(ai 2 J»- 2/3) = 0.
But aj2»»— 2/3 = 0, hence our equation is simply
^V'mimih-^) = 0.
Now 2Fi!r/3 is the couple, about the centre of inertia, produced
by the applied forces ; call it (/>, then
ImFs^ii = <{) (1)
383-] PHYSICAL APPLICATIONS. 225
381 .] Integrating once, •
I.mF'ST^ = y+/<f)di (2)
Again, as the motion considered is relative to the centre of inertia,
it must be of the nature of rotation about some axis, in general
variable. Let e denote at once the direction of, and the angular
velocity about, this axis. Then, evidently,
•a = Vetss.
Hence, the last equation may be written
'S.mzrYiTS = yJrf^dt.
Operating by S.i, we get
2m{Fem)^ = Sey + Se/<f,dt (3)
But, by operating directly by 2fSidt upon the equation (1), we get
2»?(reCT)2 =-h^ + 2fSi<i>dt (4)
(2) and (4) contain the usual four integrals of the first order.
382.] When no forces act on the body, we have ^ = 0, and
therefore '2,mw Few = y, (5)
Imir^ = ■2miFi'!:Tf = —A^, (6)
and, from (5) and (6), Sey =—Jfi (7)
One interpretation of (6) is, that the kinetic energy of rotation
remains unchanged : another is, that the vector e terminates in an
ellipsoid whose centre is the origin, and which therefore assigns
the angular velocity when the direction of the axis is given ; (7)
shews that the extremity of the instantaneous axis is always in
a plane fixed^in space.
Also, by (5), (7) is the equation of the tangent plane to (6) at
the extremity of the vector e. Hence the ellipsoid (6) rolls on the
plane (7).
From (5) and (6), we have at once, as an equation which e must
satisfy, y2 2.^ ( FimY= —k^ (2.»8sr Fivrf.
This belongs to a cone of the second degree fixed in the body. Thus
all the ordinary results regarding the motion of a rigid body under
the action of no forces, the centre of inertia being fixed, are deduced
almost intuitively : and the only difficulties to be met with in more
complex properties of such motion are those of integration, which
are inherent to the subject, and appear whatever analytical method
is employed. (Hamilton, Proc. B. I. A. 1848.)
383.] Let a be the initial position of ■nr, q the quaternion by
which the body can be at one step transferred from its initial posi-
tion to its position at time t. Then
ra- = qaq~^
Q
226 QUATEEKIONS. \.3M-
and Hamilton's equation (5) of last section becomes
or ^.mq {aS.aq~^ tq—q~^(qa? } q'"^ = y.
Let <^p = 'Si.m{a8ap—a?p), (1)
where is a self-conjugate linear and vector function, whose con-
stituent vectors are fixed in the body in its initial position. Then
the previous equation may be written
or </>(S'~^«S') = rVS'-
For simplicity let us write
r^'i = r),^
Then Hamilton's dynamical equation becomes simply
0'? = C. (3)
384.3 ^^ is ®^y *o s^^ what the new vectors r\ and ( represent.
For we may write (2) in the form
e = qm-\ \ (2')
from which it is obvious that rj is that vector in the initial position
of the body which, at time t, becomes the instantaneous axis in the
moving body. When no forces act, y is constant, and f is the
initial position of the vector which, at time t, is perpendicular to
the invariable plane.
385.] The complete solution of the problem is contained in equa-
tions (2), (3) above, and (4) of § 356*. Writing them again, we
have
qr)=M, (4)
7i = iC, (2)
0'? = f. (3)
We have only to eliminate f and »;, and we get
2q = q<f>-^q-^yq), (5)
in which q is now the only unknown ; y, if variable, being supposed
known in terms of q and t. It is hardly conceivable that any
simpler, or more easily interpretable, equation for q can be presented
* To these it is unnecessary to add
Z'g= constant,
as this constancy of Tq is proved by the form of (4). For, had Tq been variable, there
must have been a quaternion in the place of the vector i/. In &ct,
^(Tqr = 2S.qKq^{Tqf8n'=0.
387.] PHYSICAL APPLICATIONS. 227
until symbols are devised far more comprehensive in their meaning
than any we yet have.
386.] Before enfering into considerations as to the integration
of this equation, we may investigate some other consequences of
the group of equations in § 385. Thus, for instance, differentiating
(2), we have
and, eliminating q by means of (4),
yqri + 2yq = qt,C+2qC,
whence C=yCn+ i~^ yq. ;
which gives, in the case when no forces act, the forms
t=yp^-H, (6)
and (as C= ^^)
<l>ri= — F.ri<t>ri (7)
To each of these the term q~^ yq, or q~^ yjfq, must be added on the
right, if forces act.
387.] It is now desirable to examine the formation of the fanc-
tion <f). By its definition (1) we have
<l>p = 2.M (aSap — a^p),
= — 'S.maVap.
Hence —Sp(Pp = 'S,.m{Trapf,
so that — Sp<pp is the moment of inertia of the body about the
vector p, multiplied by the square of the tensor of p. Thus the
equation g^^p ^ _p^
evidently belongs to an ellipsoid, of which the radii-vectores are
inversely as the square roots of the moments of inertia about them ;
so thatj if i, j, k be taken as unit- vectors in the directions of its
axes respectively, we have
Si<j)i = — A, \
Sj<f>j=-BA (8)
Sk<t)k = -C,)
A, B, C, being the principal moments of inertia, Consequently
4>p = —{AiSip + £JSjp+ CkSip} (9)
Thus the equation (7) for rj breaks up, if w^ put
into the three following scalar equations
Aa)i+ (C— 5)q)2C»3 = 0, j
Sd}^ + {A — C) w^coj^ = 0, I
C(02 + {B — A) o>^a>2 = 0, )
Q 2
228 QUATEENIONS. [388.
which are the same as those of Euler. Only, it is to be understood
that the equations just written are not primarily to be considered
as equations of rotation. They rather expres* with reference to
fixed axes in the initial position of the body, the motion of the
extremity, toj, Ug, (1)3, of the vector corresponding to the instan-
taneous axis in the moving body. If, however, we consider tOj, Wg, cog
as standing for their values in terms of w, x,y, «: (§ 391 below), or
any other coordinates employed to refer the body to fixed axes, they
are the equations of motion.
Similar remarks apply to the equation which determines f, for if
we put f=i^^ + y^^ + ^^^^
(6) may be reduced to three scalar equations of the form
''^9'^^ = 0.
388.] Euler's equations in their usual form are easily deduced
from what precedes. For, let
whatever be p ; that is, let + represent with reference to the moving
principal axes what ^ represents with reference to the principal
axes in the initial position of the body, and we have
<t.e = q^ (q-^ iq) q'^ = q<l> (n) q'^
= qiq-' =qr{C'l>-H)q-'
= -qr{ri<t>ri)q-^
. =-V.qri<p{n)q-'^
= -r.qr,q-'^q(t){q-'^eq)q-^
which is the required expression.
But perhaps the simplest mode of obtaining this equation is to
start with Hamilton's unintegrated equation, which for the case
of no forces is simply
S.»«FisrOT = 0.
But from ot =: Vezr
we deduce «• ?= Fe^+ Vk-sr
= ore^ — e<S«CT+ Vkvr,
so that 2.«M(F'e«riS€OT — eCT^ + cr^eBr) = 0.
If we look at equation (1), and remember that ^ differs from
simply in having ot substituted for a, we see that this may be
written Fe+e + ^e = 0,
390-] PHYSICAL APPLICATIONS. 229
the equation before obtained. The first mode of arriving at it has
been given because it leads to an interesting set of transformations,
for which reason we append other two.
By (2) y = qCq-\
therefore = qq-'^.q^q-'^+q^q-'^—q^^q-'^q^'^,
or q^q-^ = iV.yVqq-'^
= Fye.
But, by the beginning of this section, and by (5) of § 382, this
is again the equation lately proved.
Perhaps, however, the following is neater. It occurs in Hamil-
ton's Elements.
By (5) of §382 +€ = y.
Hence <t>e =—<}>«=— S.w(t3- Few + ot Fenr)
= — 'Si.m'iiSesi
= — F'.f'2.m'srSe'ST
= - re4.e.
389.] However they are obtained, such equations as those of
§ 387 were shewn long ago by Euler to be integrable as follows.
letting 2fm^<i,^mjt = s,
we have j^^z =JQ^^ + (£- C) s,
with other two equations of the same form. Hence
2dt=: -
so that t is known in terms of s by an elliptic integral. Thus,
finally, tj or f may be expressed in terms of i ; and in some of the
succeeding investigations for q we shall suppose this to have been
done. It is with this integration, or an equivalent onCj that most
writers on the farther development of the subject have commenced
their investigations.
390.] By § 381, y is evidently the vector moment of momentum
of the rigid body ; and the kinetic energy is
But Sey = S.q-^eqq~^yq = SrjC
so that when no forces act
SC(l>'H=Sr]<l>r,=-AK
But, by (2), we have also
TC=Ty, or T<f>r, = Ty,
so that we have, for the equations of the cones described in the
230 QUATERNIONS. [39 1.
initial position of the body by rj and t, that is, for the cones de-
scribed in the moving body by the instantaneous axis and by the
perpendicular to the invariable plane,
This is on the supposition that y and & are constants. If forces act,
these quantities are functions of t, and the equations of the cones
then described in the body must be found by eliminating t between
the respective equations. The final results to which such a process
will lead must, of course, depend entirely upon the way in which t
is involved in these equations, and therefore no general statement
on the subject can be made.
391.] Recurring to our equations for the determination of q, and
taking first the case of no forces, we see that, if we assume tj to
have been found (as in § 389) by means of elliptic integrals, we have
to solve the equation „ .^
that is, we have to integrate a system of four other difiPerential
equations harder than the first.
Putting, as in § 3 8 7, n = icOj^ +j\ + kw^ ,
where Wj, Wg, W3 are supposed to be known functions of t, and
q = w+ico + jy + kz,
... , . \ ,, dm dx du dz
this system IS -di = ^ = y = Y ~ 'Z'
* To get an idea of the nature of this equation, let us integrate it on the supposi-
tion that ij is a constcmt vector. By differentiation and substitution, we get
Hence „_ «,.«=. ^^ * j_ n =i- ^^ t
g= ^icos — « + QsSin^ t.
Substituting in the given equation we have
2^ C^ e, sin 2l e +& cos ^ «) = («, cos ^ «+ e, sin ^ ^j-J-
Hence Tiy.Ga = Q, 1;,
which are virtually the same equation, and thus.
And the interpretation of 2 ( ) q~^ will obviously then be a rotation about ij through
the angle tTrj, together with any other arbitrary rotation whatever. Thus any posi-
tion whatever may be taken as the initial one of the body, and Q, ( ) Q,-» brings it
to its required position at time < = 0.
3 9 2. J PHYSICAL APPLICATIONS. 231
m
where ^= — <o,a;— Wgy — ojj^,
X= Wj^W + tBg^ — a^i^,
^= w^w + a^se — ooi^l
or, as suggested by Cayley to bring out the skew symmetry,
X= . (ja^y — oi^z + ai-^w,
T=.—m^x . + a-j^z + (o^w,
Z ■=■ oj^a; — Wj^y . -^m^w,
W ■=—<it>-^X — ai2,y — <«>3« .
Here, of course^ one integral is
w^ +(xi'^ +^^+z^ = constant.
It may suffice thus to have alluded to a possible mode of solution,
which, ^except for very simple values of ri, involves very great diffi-
culties. The quaternion solution, when rj is of constant length and
revolves uniformly in a right cone, will be given later.
392.] If, on the other hand, we eliminate t], we have to inte-
grate S^~^ir^72)=^i'
so that one integration theoretically suffices. But, in consequence
of the present imperfect development of the quaternion calculus, the
only known method of effecting this is to reduce the quaternion
equation to a set of four ordinary differential equations of the first
order. It may be interesting to form these equations.
Put q = w+iai + jy + iz,
Y = ia+Jb + ^o,
then, by ordinary quaternion multiplication, we easily reduce the
given equation to the following set :
di d/w dx dy dz
where
W= — x'^—y3&—ze. or X= . yC— «13+wa,
x= wa+^ffi— ^B r=— ««[; . +z^+wii,
T= w'Q+z%—ui!<S; z= !JBi&—y% . +w(i::,
and
a = -J [a (w^— a;'' —y^ —z^) + 2a? {m + hy -\-ez) + 2w {bz—cy)'],
33 = -^ [5 {w^ —afi —y"^ —z^) + ly {ax + by + cz) + 2w {cx—az)'],
a: = -^ [c {w^ —x^ —y^—z^) + 2z{ax+6y + cz) + 2w {ay-^bx)],
232 QUATERNIONS. [393«
JF, X, Y, Zare thus homogeneom functions of w, x, y, z of the third
degree.
Perhaps the simplest way of obtaining these equations is to trans-
late the group of § 385 into w, x, y, z at once, instead of using the
equation from which f and r\ are eliminated.
We thus see that ^ ^ *a+yi8 +/^ffi.
One obvious integral of these equations ought to be
vfi + x"^ +y^ +z^ = constant,
which has been assumed all along. In fact, we see at once that
wV+xX+yY+zZ=
identically, which leads to the above integral.
These equations appear to be worthy of attention, partly because
of the homogeneity of the denominators W, X, T, Z, but particularly
as they afford (what does not appear to have been sought) the means
of solving this celebrated problem at one step, that is, without the
previous integration of Euler's equations (§ 387).
A set of equations identical with these^ but not in a homogeneous
form (being expressed, in fact, in terms of k, \, |u, v of § 359, instead
of 10, x,y, z), is given by Cayley {Gamb. and Bub. Math. Journal,
vol. i. 1846), and completely integrated (in the sense of being re-
duced to quadratures) by assuming Euler's equations to have been
previously integrated. (Compare § 391.)
Cayley's method may be even more easily applied to the above
equations than to his own ; and I therefore leave this part of the
development to the reader, who will at once see (as in § 391) that
%, 38, ffi correspond to coi, Wg, tag of the rj type, § 387.
393.J It may be well to notice, in connection with the formulae
for direction cosines in § 359 above, that we may write
% = --j\a{:w'^-\-x^—y'''—z^)-^il{xy + 'wz)-\-'ic{pz—wy)'],
38 = -^\2a{xy — wz)-{h(vP-—x'^->ry'^—z''-)-\-1c{yz-\-wxy\,
(!t = -p^[2a(xz + wy) + 2b {yz—wx) + c {w^ —x^ —y"^ + z^)'].
These expressions may be considerably simplified by the usual
assumption, that one of the fixed unit- vectors {i suppose) is perpen-
dicular to the invariable plane, which amounts to assigning defi-
nitely the initial position of one line in the body ; and which gives
the relations 5—0 c =
396.] PHYSICAL APPLICATIOlSrS. . 233
•
394.] Wlieii forces act, y is variable, and the quantities a, h, c
will in general involve all the variables w, x, y, z, t, so that the
equations of last section become much more complicated. The type,
however, remains the same if y involves t only ; if it involve q we
must differentiate the equation, put in the form
y=2q(l>{q-^q)q-^,
and we thus easily obtain the differential equation of the second
order ^ = iV.qct) (q-^ q) q-^ + 2 qcj) {F. q-^q) q-^ ;
if we recollect that, because q~^q is a vector, we have
8.q-^=(3-^qf.
Though remarkably simple, this formula, in the present state of
the development of quaternions, must be looked on as intractable,
except in certain very particular cases.
395.] Another mode of attacking the problem, at first sight
entirely different from that in § 383, but in reality identical with
it, is to seek the linear and vector function which expresses the
Homogeneous Strain which the body must undergo to pass from its
initial position to its position at time t.
Let -ST = xfflj
a being (as in § 383) the initial position of a vector of the body,
■ST its position at time t. In this case x i^ ^ linear and vector
function. (See § 360.)
Then, obviously, we have, ^-^ being the vector of some other point,,
which had initially the value a^,
Siss'ST^ = S.\a)(a.i = Saa^,
(a particular case of which is
T'ST = ^xa = Ta)
and Fototj = J^-x^X^^i = x^"«i'
These are necessary properties of the strain-function x, depending
on the fact that in the present application the system is rigid.
396.] The kinematical equation
CT = Few
becomes Xa = F. exa
(the function x being formed from x by the differentiation of its
constituents with respect to t).
Hamilton's kinetic equation
S.warFera- = y,
becomes 'Si.mxaF.exa = y.
234 QUATERNIONS. [39 7>
This may be written
2.««(xaiS'.€xa— eo^) = V)
or I,.m{aS.ax'f-x~^e.a^) = X~^Y>
where x' is the conjugate of x-
But, because '^•X'^X'h. = '^""u
we have Saa^ = S.ax'xa^,
whatever be a and a^, so that
X = X ^•
Hence 2.m{w8.ax~^e—x~^e.a^) = x~V,
or, by §383, ^^-i^^^-iy^
397.] Thus we have, as the analogues of the equations in
§§ 383, 384, ^-1^ ^ ^^
x-V = C,
and the former result x" = ^' «X°
becomes X** =^'X'7X'* = X^**-
This is our equation to determine X) V being supposed known.
To find rj we may remark that
<f>l = C,
and C = X~V-
But XX~^« = a.
so that XX~^« + XX~'« = 0.
Hence f=-X~^XX~V
= -r.r,x-^Y=J'Cv=^-C4>-'C
or </>^ = — Ftj^tj.
These are the equations we obtained before. Having found rj
from the last we have to find x from the condition
X-^Xa=Fria.
398.] We might, however, have eliminated ?j so as to obtain an
equation containing x a^lone, and corresponding to that of § 385.
For this purpose we have
jj = ,^-if= ^-^x"^y>
so that, finally, X~^X'"' = ^- 't>~^ X~''>"»>
or X~^ « = ^' X" ^ <'0~^X~^y'
which may easily be formed from the preceding equation by putting
X~^a for a, and attending to the value of x"^ given in last section.
40I.J PHYSICAL APPLICATIONS. 235
•
399.] We have given this process, though really a disguised form
of that in §§ 383, 385, and though the final equations to which
it leads are not quite so easily attacked in the way of integration as
those there arrived at, mainly to shew how free a use we can make
of symholic functional operators in quaternions without risk of
error. It would be very interesting, however, to have the problem
worked out afresh from this point of view by the help of the old
analytical methods : as several new forms of long-known equations,
and some useful transformations, would certainly be obtained.
400.] As a verification, let us now try to pass from the final
equation, in x alone, of § 398 to that of § 385 in ^ alone.
We have, obviously,
OT = qa£r^ = X«.
which gives the relation between q and x-
[It shews, for instance, that, as
yS.^Xa = -S.ax'A
while 'S-zSxa = S.^qaq-'^ = S.aq-^^q,
we have x'/3 = T'^Pi^
and therefore that xx'i^ = id'^Pi)^^ = i^,
or x' = X~^j ^s above.]
Difierentiating, we have
qaq^'^—qaq'^qq'^ = x«-
Hence X'^X" = S'~^?<*~"2^*?
= 2r.r{q-'^q)a.
Also ^~^X~V = ^"^(^^V?).
so that the equation of § 398 becomes
2r.r{q-'^q)a= V. (^-^ {q-^7q) a,
or, as a may have any value whatever,
2r.q'^q = ^-Hq-^yq),
which, if we put Tq = constant
as was originally assumed, may be written
2q = q<l>-\q-^yq),
as in § 385.
401.] To form the equation for Precession and Nutation,. Let o-
be the vector, from the centre of inertia of the earth, to a particle
m of its mass : and let p be the vector of the disturbing body, whose
mass is M. The vector-couple produced is evidently
236 QUATERNIONS. [4OI.
= M^.
mVap
no farther terms being necessary, since =- is always small in the
actual cases presented in nature. But, because o- is measured from
the centre of inertia, S.?»o-= 0.
Also, as in § 383, <^p = 2.«! {aScrp—tr'^p).
Thus the vector-couple required is
Referred to coordinates moving with the body, ^ becomes 4> as in
§ 388, and § 388 gives
♦e=y=3Jf/^
P^Pdt.
Simplifying the value of <|> by assuming that the earth has two
principal axes of equal moment of inertia, we have
Bf—{A—B)aSaf = vector-constant + ZM{A—B) / ^g °^
dt.
This gives Sat = const. = i2,
whence e = — i2a -|- act,
so that, finally,
BVad-Aaa = ^{A-B)rap8ap.
The most striking peculiarity of this equation is that Reform of
the solution is entirely changed, not modified as in ordinary cases
of disturbed motion, according to the nature of the value of p.
Thus, when the right-hand side vanishes, we have an equation
which, in the case of the earth, would represent the rolling of a
cone fixed in the earth on one fixed in space, the angles of both
being exceedingly small.
If p be finite, but constant, we have a case nearly the same as
that of a top, the axis on the whole revolving conically about p.
405. j PHYSICAL APPLICATIONS. 237
But if we assume the expr *sion
p = r{Jeosmt + k sin mt)j
(which represents a circular orbit described with uniform velocity,)
a revolves on the whole conically about the vector i, perpendicular
to the plane in which p lies. {Trans. B, 8. E., 1868-9.)
402.] To form the eq%iation of motion of a simple 'pendulum,
taking account of the eartVs rotation. Let a be the vector (from
the earth's centre) of the point of suspension, X its inclination to
the plane of the equator, a the earth's radius drawn to that point ;
and let the unit-vectors i, j, h be fixed in space, so that i is parallel
to the earth's axis of rotation ; then, if m be the angular velocity
of that rotation
a = « p sin A + (/ cos 01^ + ^ sin ad) cos A] (1)
This gives a = a o) ( —j sin tu^ + A cos mf) cos \
^ = inYia ...(2)
Similarly a = m Yia = — o)^ (a — ai sin A) (3)
403.] Let p be the vector of the bob m referred to the point of
suspension, R the tension of the string, then if oj be the direction
ofpuregravity m{d + p) =-mgUay-BUp, (4)
which may be written
rpd+rpp = ^ja,p (5)
To this must be added, since r (the length of the string) is constant,
Tp = r, (6),
and the equations of motion are complete.
404.] These two equations (5) and (6) contain every possible case
of the motion, from the most infinitesimal oscillations to the most
rapid rotation about the point of suspension, so that it is necessary
to adapt different processes for their solution in different cases.
We take here only the ordinary Foucault case, to the degree of
approximation usually given.
405.] Here we neglect terms involving m^. Thus we write
a = 0,
and we write a for Oj , as the difference depends upon the ellipticity
of the earth. Also, attending to this, we have
T
p= — -a + i!T, (7)
whereby (by (6)) xSoot = 0, (8)
and terms of the order ot^ are neglected.
238 QUATERNIONS. [405.
With (7), (5) becomes
— — Vwss = — Foot ;
a a
so thatj if we write -■=•«?, (9)
we have FaC* + w^ot) = (10)
Now, the two vectors ai— asia\ and Via
have, as is easily seen, equal tensors ; the first is parallel to the line
drawn horizontally northwards from the point of suspension, the
second horizontally eastwards.
Let, therefore, w = «;(«»- o sin A) +j^ria, (11)
which {x and y being very small) is consistent with (6).
From this we have (employing (2) and (3), and omitting a?)
•is = cb {ai— asinX) + yFia—xm ainXFia— yo) {a— ai sin \),
a =z x{ai — aaiaK)+ifFia—2dia>BmKFia—2ya){a—aiBia\).
With this (10) becomes
Fa[ai(aJ— a sin \) + yFia—2xoi s\n\Fia—2ym{a—ai sin \)
+ n^x{ai—asm\)^n^yFia] = 0,
or, if we note that F. a Fia = a{ai—a sin \),
(^—x—2ya>smk—n,^x)aFia + {t/ — 2ii;a)8in.k + n'^y)a(ai—asm\) = 0.
This gives at once x + n^x+ 2a>jfsm\ = 0,
y+n^y—2a>xsin\
which are the equations usually obtained ; and of which the solution
is as follows : —
If we transform to a set of axes revolving in the horizontal plane
at the point of suspension, the direction of motion being from the
positive (northward) axis of x to the positive (eastward) axis of y,
with angular velocity ii, so that
a; = f cos Slt—r) sin Sit,
^ = f sin Qft + t) cos 12 1,
and omit the terms in D? and in w 12 (a process justified by the
results, see equation (15)), we have
({+«^0 cos Q,t-(ij + n^ri) sin Q.t-2^ {il—co sinX) = 0, )
So that, if we put il = oism\, (15)
we have simply f +*^£ = 0, )
ij + n''r, = 0j ^^"^
the usual equations of elliptic motion about a centre of force in the
centre of the ellipse. (Proc. E. S. K, 1869.)
=::} <->
';} <"'
407.-] PHYSICAL APPLICATIONS. 239
406.] To construct a reflecHkig surface from which rays, emitted
from a point, shall after reflection diverge uniformly, hut horizontally.
Using the ordinary property of a reflecting surface, we easily
obtain the equation
S.dp{^±^% = Q.
By Hamilton's grand Theory of Systems of Bays, we at once write
down the second form
Tp—T(fi+aFap) = constant.
The connection between these is easily shewn thus. Let ot and
T be any two vectors whose tensors are equal, then
whence, to a scalar factor ^re*, we have
\i T + 'S!-
=
T
Hence, putting w = C/'(/3 + aVap) and r = Up, we have from the first
equation above
S.dplUp+ Ui^ + aVap)'] = 0.
But d(p + aFap) = aVadp =—dp—aSadp,
and S.a(fi + a Vap) = .0,
so that we have finally
S.dpUp-S.d{^ + aFap)U{^ + arap) = 0,
which is the differential of the second equation above. A curious
particular case is a parabolic cylinder, as may be easily seen geo-
metrically. The general surface has a parabolic section in the plane
of a, y3 ; and a hyperbolic section in the plane of /3, a0.
It is easy to see that this is but a single case of a large class of
integrable scalar functions, whose general type is
S.dp(^'p = 0,
the equation of the reflecting surface ; while
8{<T—p)dcT —
is the equation of the surface of the reflected wave : the integral of
the former being, by the help of the latter, at once obtained in the
form Tp + ^(a—p) = constant*.
407.] We next take Fresnel's Theory of DouMe Refraction, but
* Proe. R. S. E., 1870-71.
240 QUATERNIONS. [408.
merely for the purpose of shewing how quaternions simpHfy the
processes required, and in no way to discuss the plausibility of the
physical assumptions.
Let tzT he the vector displacement of a portion of the ether, with
the condition ^2 __i /j\
the force of restitution, on Fresnel's assumption, is
tiflHSvar + b^jSj':!T + c^kSkin) = t<fm,
using the notation of Chapter V. Here the function <^ is obviously
self-conjugate, a^, b^, c^ are optical constants depending on the
crystalline medium, and on the colour of the lightj and may be
considered as given.
Fresnel's second assumption is that the ether is incompressible,
or that vibrations normal to a wave front are inadmissible. If, then,
a be the unit normal to a plane wave in the crystal, we have of
course a^=-\, (2)
and Six's! = 0; (3)
but, and in addition, we have
■s!~^ Vtz^Ts II a,
or S.aTu^ = (4)
This equation (4) is the embodiment of Fresnel's second assumption,
but it may evidently be read as meaning, the normal to the front, the
direction, of vibration, and that of the force of restitution are in one
plane,
408.] Equations (3) and (4), if satisfied by -m, are also satisfied
by Tsa, so that the plane (3) intersects the cone (4). in two lines
at right angles to each, other. That is, for any given wave front
there are two directions of vibration, and they are perpendicular to each
other.
409.] The square of the normal velocity of propagation of a plane
wave is proportional to the ratio of the resolved part of the force of
restitution in the direction of vibration, to the amount of displace-
ment, hence j;2 = S-as^Tn,
Hence Fresnel's Wave-surface is the envelop of the plane
Sap ^ i\/Sm<^, (5)
with the conditions vt^ = — \, (1)
a''=-l, (2)
Sour =0, (3)
S. aiJ7<l)'ar = (4)
409-] PHYSICAL APPLICATIONS. 241
Formidable as this problem appears, it is easy enough. From (3)
and (4) we get at onee^
Henee^ operating by S. ct,
— CO ^ — S'STcfyar = — v^.
Therefore ((jb + »2) ^ = _ a^ac^^-,
and S.a {(j) + v^)-^ a = (6)
In passing, we may remark that tMs equation gives the normal velo-
cities of the two rays whose fronts are perpendicular to a. In Cartesian
coordinates it is the well-known equation
P wfi rfi _
a^—v^ ■*" P3^2 + ^2~^ = °-
By this elimination of or, our equations are reduced to
S.a{(i> + v^)-'^a= 0, (6)
V zzz-Sap, (5)
a^ =-1 (2)
They give at once, by § 309,
{ct> + v^)-^a + vpSa{cj>-{-v^)-^a = ha.
Operating by S.a we have
v^Sa{<tj + v^)-^a = h.
Substituting for h, and remarking that
Sa{(t> + v^)-^a =-T^{(j> + t)2)-i a,
because <^ is self-conjugate, we have
/J . 2\-i va — p
p^ + v^
This gives at once, by rearrangement,
^{(l> + v^)-^a = {<t>-p^)-Y
Hence {<t>-p^)-^P = ^^^ ■
Operating by S.p on this equation we have
Sp{<P-p')-^p = -l, (7)
which is the required eqjflation.
[It will be a good exercise for.the student to translate the last
ten formulae into Cartesian coordinates. He will thus reproduce
almost exactly the steps by which Archibald Smith * first arrived
at a simple and symmetrical mode of .effecting the elimination. Yet,
as we shall presently see, the above process is far from being the
shortest and easiest to which quaternions conduct us.J
* Cambridge Phil. Trans., 1835.
242 QUATEENIONS. [4IO.
410.] The Cartesian form of the equation (7) is not the usual
one. It is, of course,
aj2 yi g^
But write (7) .in the form
and we have the usual expression
a2^2 ^2 ,,2 „2,2
I 7,2 „2 T .!! ..•>. "
ast quaternion eqi
forms
This last quaternion equation can also be put into either of the new
or 2'(p-2-,^-i)-4p = 0.
411.] By applying the results of §§ 171, 172 we may introduce
a multitude of new forms. We must confine ourselves to the most
simple ; but the student may easily investigate others by a process
precisely similar to that which follows.
Writing the equation of the wave as
where we have g = — p~^,
we see that it may be changed to
if mSp<f>p = ffkp^ ■=—h.
Thus the new form is ^
Sp{(j)-^—mSp(l)p)-^p = (1)
Here m = -^^^ , 8p^p = a^ap' + V^y"^ + c^z^,
and the equation of the wave in Cartesian coordinates is, putting
' ^ + -...« ... = 0.
412.] By means of equation (1) of last section we may easily
prove Pliicker's Theorem, The Wave-Surface is its own reciprocal with
resjieci to the ellipsoid, whose equation is
Sp^^p = —7— •
41 3-] PHYSICAL APPLICATIONS. 243
The equation of the plane of contact of tangents to this surface from
the point whose vector is p is
iSWd)* p = —, —
The reciprocal of this platie, with respect to the unit-sphere about
the origin, has therefore a vector cr where
a = \/m,(ji^ p.
Hence p = —t— (b~^a;
and when this is substituted in the equation of the wave we have
for the reciprocal (with respect to the unit-sphere) of the reciprocal
of the wave with respect to the above ellipsoid,
S.cr (^ - — Sacj)-^ 0-) 0- = 0.
This differs from the equation (1) of last section solely in having
(p~^ instead of (f>, and (consistently with this) — instead of m. Hence
it represents the index-surface. The required reciprocal of the wave
with reference to the ellipsoid is therefore the wave itself.
413.J Hamilton has given a remarkably simple investigation of
the form of the equation of the wave-surface, in his Elements, p. 736,
which the reader may consult with advantage. The following is
essentially the same, but several steps of the process, which a skilled
analyst would not require to write down, are retained for the benefit
of the learner.
Let %= — 1 (1)
be the equation of any tangent plane to the wave^ i.e. of any wave-
front. Then /u is the vector of wave-slowuess, and the normal
velocity of propagation is therefore -=p- . Hence, if isr be the vector
direction of displacement, ju~^«r is the effective component of the
force of restitution. Hence, ^w denoting the whole force of re-
stitution, we have ^'sr—pr^'oi || p.,
or -m II {4>—ijr^)-^p.,
and, as ss is in the plane of the wave-front,
Sp.'d = 0,
or SiJi.{(f)-p.-'')-^iJ. = (2)
This is, in reality, equation (6) of § 409. It appears here, how-
ever, as the equation of the Index-Surface, the polar reciprocal of
E %
244 QUATERNIONS. [4 1 4.
the wave with respect to a unit-sphere about the origin. Of course
the optical part of the problem is now solved, all that remains being
the geometrical process of § 3 1 1 .
414.] Equation (2) of last section may be at once transformed,
by the process of § 410, into
5f.((^2-<|,-i)-V=i.
Let us employ an auxiliary vector
whence ij,= (jx'^—(J)-^)t (1)
The equation now becomes
Sh.t=1, (2)
or, by (1), y?T-^-S!r4r'^T = 1 .- (3)
Differentiating (3), subtract its half from the result obtained by
operating with S.t on the differential of (1). The remainder is
T'^Sixdn—STdjj. = 0.
But we have also (§311) Spdix = 0,
and therefore xp = jxt^—t,
where a; is a scalar.
This equation, with (2), shews that
Stp = (4)
Hence, operating on it by S.p, we have by (1) of last section
xp^ = — r^,
and therefore p~^ =— /x + r"^.
This gives p~^ = ij,^ — t~'^.
Substituting from these equations in (1) above, it becomes
or r = ((^~^— p~^)~^p~^.
Finally, we have for the required equation, by (4),
^p-i(<^-i_p-2)-ip-i = 0,
or, by a transformation already employed,
Sp{cf>-p^)-^p=-l.
415,] It may assist the student in the practice of quaternion
analysis, which is our main object, if we give a few of these invest-
igations by a somewhat varied process.
Thus, in § 407, let us write as in § 168,
aHSv^Jfl^jSj-ss^c^hSk^ = yxS/OT + Z/SW-yOT.
We have, by the same processes as in § 407,
S.VTaX'Si/t!r + S.'maix'Sk''!!T = 0.
4I7.J
PHYSICAL APPLICATIONS,
245
(1)
This may be written, so flr as the generating lines we require are
concerned,
since -sra is a vector.
Or we may write
S.[l,'V.'7T\'-S!a = = /S./yl'OTX'OTa.
Equations (1) denote two cones of the second order which pass
through the intersections of (3) and (4) of § 407. Hence their in-
tersections are the directions of vibration.
416.] By (1) we have
S.T!TX.''sraix'= 0.
Hence ■nrX'tn-j a, \i.' are coplanar ; and, as tn- is perpendicular to a, it
is equally inclined to Vk'a and Fix a.
For, i£ L, M, A be the projections of k', f/, a on the unit
sphere, £C the g-reat circle whose
pole is A, we are to find for the
projections of the values of w^ on
the sphere points P and P', such
that if LF be produced till
Q may lie on the great circle AM.
Hence, evidently,
CP = PB,
and C^F=rB;
which proves the proposition, since
the projections of Vk'a and Vj/a on the sphere are points b and
c in BC, distant by quadrants from C and B respectively.
417.] Or thus, Svra = Q,
S.srV.ak'-snx — 0,
therefore as'sr = F. a K ak''as-ii,
= - r. W/ -aSaF. W/x'.
Hence {Sk'ix-a;) ot = (X' + aSak') ^/x'w + {/ + aSaf/) Sk'w.
Operate by S.k', and we have
(x + Sk'aSi/a) Sk'^ = [X'^ a^-S'^ X'a] -S/xV
= Si/^T'^Fk'a.
Hence by symmetry,
^''''^T^Fk'a=f^T^Fi/a,
Sk'^a
Sjjfz
246 QUATERNIONS. [4 1 8.
"'' T7k'a - TFi/a - '
and as fco = 0,
^=U{Ur\'a±Ur,jfa).
418.] The optical interpretation of the common result of the
last two sections is that the planes of polarization of the two rays
whose wave-fronts are parallel, iisect the angles contained hy planes
passing through the normal to the wave-front and the vectors (optic
axes) A'j fx'.
419.] As in § 409, the normal velocity is given by
v^ ^SsTCJysr = 2SX.''aSf/tsy-p'^^
= / + ;
{T+8).r\'aF/a
[This transformation, effected by means of the value of or in
§ 417, is left to the reader.]
HencCj if w^, v^ be the velocities of the two waves whose normal
is a, „2 _ ^.| ^ 2 T. r\'a r/a
oc sin K'a sin ju'o.
That is, the difference of the squares of the velocities of the two waves
varies as the product of the sines of the angles between the normal to
the wave-front and the optic axes (A', \j.').
420.] We have, obviously.
Hence v^=p'^^ {T± S). VK'a Ff/a.
The equation of the index surface, for which
Tp = -, Up = a,
V
is therefore 1 = -p'p^ + {T±S). Fx'p Fpfp.
This will, of course, become the equation of the reciprocal of the
index-surface, i.e. the wave-surface, if we put for the function ^ its
reciprocal : i. e. if in the values of A', p.', p' we put - , y- , - for
a, b, c respectively. We have then, and indeed it might have been
deduced even more simply as a transformation of § 409 (7),
\ = -pp^i;-{T±S).F\pFp.p,
as another form of the equation of Fresnel's wave.
424-J PHYSICAL APPLICATIONS. 247
If we employ the i, k transformation of § 1 2 1, this may be written,
as the student may easily prove, in the form
421. J We may now, in furtherance of our object, which is to
give varied examples of quaternions, not complete treatment of any
one subject, proceed to deduce some of the properties of the wave-
surface from the diflFerent forms of its equation which we have
given.
422.] Fresnel's construction of the wave hy points.
From § 273 (4) we see at once that the lengths of the principal
semidiameters of the central section of the ellipsoid
Sp<^-^p = 1,
by the plane Sap = 0,
are determined by the equation
6'.a(<^-i-p-2)-ia=0.
If these lengths be laid off along a, the central perpendicular to the
cutting plane, their extremities lie on a surface for which a = Vp^
and Tp has values determined by the equation.
Hence the equation of the locus is
as in §§409, 414. ^P (r^-P'^V = 0,
Of course the index-surface is derived from the reciprocal ellip-
soid Sp>^p = 1
by the same construction.
423.] Again, in the equation
\=-pp':^{T±S).YKpTp.p,
suppose VKp = 0, or F/xp = 0,
we obviously have
U\ , Up.
P = ±—7= or p = ±—=>
vj) vp
and there are therefore four singular points.
To find the nature of the surface near these points put
UK
P = V^ + ^'
where Tsr is very smallj and reject terms above the first order in
Ttsr. The equation of the wave becomes, in the neighbourhood of
the singular point,
2^35^^ + /S.OT r. X VXp. = ±T. TAot FX/x,
which belongs to a cone of the second order.
424.] From the similarity of its equation to that of the wave, it
248 QUATERNIONS. [425.
is obvious that the index-surface also has four conical cusps. As
an infinite number of tangent planes can be drawn at such a point,
the reciprocal surface must be capable of being touched by a plane
at an infinite number of points ; so that the wave-surface has four
tangent planes which touch it along ridges.
To find their form, let us employ the last form of equation of the
wave in § 420. If we put
Trip=TrKp, (1)
we have the equation of a cone of the second degree. It meets the
wave at its intersections with the planes
S{l-K)p=+{K^-i^) (2)
Now the wave-surface is touched by these planes, because we cannot
have the quantity on the first side of this equation greater in abso-
lute magnitude than that on the second, so long as p satisfies the
equation of the wave.
That the curves of contact are circles appears at once firom (1)
and (2), for they give in combination
p2 = +5(t + K)p, (3)
the equations of two spheres on which the curves in question are
situated.
The diameter of this circular ridge is
[Simple as these processes are, the student will find on trial that
the equation Sp{<f>~''-—p~^)~'^p = 0,
gives the results quite as simply. For we have only to examine
the eases in which — p"^ has the value of one of the roots of the
symbolical cubic in (^"^. In the present case Tp = b is the only one
which requires to be studied.]
425.] By § 41 3, we see that the auxiliary vector of the succeed-
ing section, viz.
is parallel to the direction of the force of restitution, 0in-. Hence,
as Hamilton has shewn, the equation of the wave, in the form
Srp = 0,
(4) of §414, indicates that fJie direction of the force of restitution is
perpendicular to the ray.
Again, as for any one versor of a vector of the wave there are two
values of the tensor, which are found from the equation
429-j PHYSICAL APPLICATIONS. 249
we see by § 422 that the lines of vibration for a given plane front
are parallel to the axes of any section of the ellipsoid,
S.p(t>-^p = 1
made hy a plane parallel to the front ; or to the tangents to the lines
of curvature at a point where the tangent plane is parallel to the wave-
front.
426.] Again, a curve which is drawn on the loave-surface so as to'
touch at each point the corresponding line of vibration has
Hence S(ppdp = 0, or Sp^p = C,
so that such, curves are the intersections of the wave with a series
of ellipsoids concentric with it.
427,] For curves cutting at right angles the lines. of vibration we
have dp II Fp(j)-^ ((/)-! -p-^)-V
\\rp{cj,-p^)-^p.
Hence Spdp = 0, or Tp = C,
so that the curves in question lie on concentric spheres.
They are also spherical conies, because where
Tp = C
the equation of the wave becomes
the equation of a cyclic cone, whose vertex is at the common centre
of the sphere and the wave-surface, and which cuts them in their
curve of intersection. (Quarterly Math. Journal, 1859.)
428.] As another example we take the case of the action of
electric currents on one another or on magnets; and the mutual
action of permanent magnets.
A comparison between the processes we employ and those of
Ampere {Theorie des Phenomenes Mectrodynamiques, ^c, many of
which are well given by Murphy in his Electricity) will at once
shew how much is gained in simplicity and directness by the use of
quaternions.
The same gain in simplicity will be noticed in the investigations
of the mutual effects of permanent magnets, where the resultant
forces and couples are at once introduced in their most natural and
direct forms.
429.] Ampere's experimental laws may be stated as follows :
I. Equal and opposite currents in the same conductor produce
equal and opposite effects on other conductors : whence it follows
250 QUATERNIONS. [430.
that an element of one current has no effect on an element of an-
other which lies in the plane bisecting the former at right angles.
II. The effect of a conductor bent or twisted in any manner is
equivalent to that of a straight one, provided that the two are
traversed by equal currents, and the former nearly coincides with
the latter.
III. No closed circuit can set in motion an element of 'a circular
conductor about an axis through the centre of the circle and per-
pendicular to its plane.
IV. In similar systems traversed by equal currents the forces are
equal.
To these we add the assumption that the action between two
elements of currents is in the straight line joining them : and two
others, viz. that the effect of any element of a current on another is
directly as the product of the strengths of the currents^ and of the
lengths of the elements.
430.] Let there be two closed currents whose strengths are a
and a^; let a, Oj be elements of these, a being the vector joining
their middle points. Then the effect of a on oj must, when resolved
along Oj, be a complete differential with respect to a (i.e. with respect
to the three independent variables involved in a), since the total
resolved effect of the closed circuit of which a' is an element is zero
by III.
Also by I, II, the effect is a function of Ta, Saa, Saa^, and 8a a^,
since these are suflScient to resolve a and Oj into elements parallel
and perpendicular to each other and to a. Hence the mutual effect
is aa-JJaf{Ta, Saa, Saa^, Si/aj),
and the resolved effect parallel to a^ is
aiZj SUai TJaf.
Also, that action and reaction may be equal in absolute magnitude,
ymust be symmetrical in Sao! and Saa-^. Again, d (as differential
of a) can enter only to the first power, and must appear in each term
of/.
Hence f^ASaa-^-\-^SaaSaa^.
But, by .IV, this must be independent of the dimensions of the
system. Hence J is of — 2 and ^ of — 4 dimensions in Ta. There-
^""■^ ^ {ASaa^Sda^ + BSaa'S^aa^}
is a complete differential, with respect to a, if da = a. Let
433-] PHYSICAL APPLICATIONS. 251
where C is a constant deperaing on the units employed, therefore
=.-=r; baa,
2Ta^ ~ Ta
and the resolved effect
Gaa^ S^aa^ Saa, „„ , , » o /o ^
"^ W^ IhF "^ 1 Ta Ta^ ^ ~ "i +^ 1^
= Caa^ „ y,^g {S. Vaa' Faa^-{-\ Saa'Saa^ .
The factor in brackets is evidently proportional in the ordinary
notation to sin 6 sin 6'cos ia — \ cos 6 cos 6'.
431.] Thus the whole force is
Caa-^a , S^aa-^ _ Caa■^^a , S'^aa'
as we should expect, d-^a being = a^. [This may easily be trans-
formed into 2Caa,Ua
which is the quaternion expression for Ampere's well-known form.]
432.] The whole effect on Oj of the closed circuit, of which a is
an element, is therefore
Cfeffj C a JSaa-^^
H f a
J Saa^
2 J Saa, Ta
3
between proper limits. As the integrated part is the same at both
limits, the effect is
^•^^IF a I, a f^°-^' fdUa
- V-^"^^' ''^''' ^=J'T^=J-^'
and depends on the form of the closed circuit.
433.] This vector ^, which is of great importance in the whole
theory of the effects of closed or indefinitely extended circuits, cor-
responds to the line which is called by Ampere " direcfrice de V action
electrodynamique" It has a definite value at each point of space,
independent of the existence of any other current.
Consider the circuit a polygon whose sides are indefinitely small;
join its angular points with any assumed point, erect at the latter,
perpendicular to the plane of each elementary triangle so formed, a
(■>
vector whose length is - > where to is the vertical angle of the tri-
252 QUATERNIONS. [434-
angle and r the length of one of the containing sides ; the sum of
such vectors is the " directrice" at the assumed point.
434.] The meve/orm of the result of § 432 shews at once that
if the element Oj he turned about its middle point, the direction of the
resultant action is confined to the plane whose normal is j3.
Suppose that the element Oj is forced to remain perpendicular to
some given vector 6, we have
Soj^b = 0,
and the whole action in its plane of motion is proportional to
Tr.bFa^^.
But r.bra^li=-a^S^b.
Hence the action is evidently constant for all possible positions
of a^ ; or
The effect of any system of closed currents on an element of a con-
ductor which is restricted to a given plane is {in that plane) independent
of the direction of the element.
435.] Let the closed current be plane and very small. Let e
(where Tt =■ 1 ) be its normal, and let y be the vector of any point
within it (as the centre of inertia of its area) ; the middle point of
oj being the origin of vectors.
Let a = y + p; therefore a'= p,
and .-/•^""-/• ^(y + P) /
and P-J Ta?-J T(y + pY
^/r(..-,).'{l+^^-^
to a sufficient approximation.
Now (between limits) fVpp'= 2Ae,
where A is the area of the closed circuit.
Also generally
fVyp'Syp =^\{SypVyp^y7.yfVpp')
= (between limits) AyVye.
Hence for this case
A , 3yFye>.
^=TyS{^' + -^)
A ( 3y% x
- Ty^\ '^ Ty"^ )'
436.] If, instead of one small plane closed current, there be a
series of such, of equal area, disposed regularly in a tubular form,
let X be the distance between two consecutive currents measured
along the axis of the tube; then, putting y'= xs, we have for the
whole effect of such a set of currents on a^
438. j PHYSICAL APPLICATIONS. 253
g-^^«i V. fry J. ^y^yY' \
CAaa.^ Va■^y ,. , t -j. x
= — - — - „ 3 (between proper limits).
If the axis of the tubular arrangement be a closed curve this will
evidently vanish. Hence a closed solenoid exerts no influence on an
element of a conductor. The same is evidently true if the solenoid he
indefinite in both directions.
If the axis extend to infinity in one direction, and y^ be the
vector of the other extremity, the effect is
CAaa^ VoiVo
and is thevefove perpendicular to the element and to the line joining it
with the extremity of the solenoid. It is evidently inversely as Ty'%
and directly as the sine of the angle contained letmeen the direction of
the element and that of the line joining the latter with the extremity of
the solenoid. It is also inversely as x, and therefore directly as the
number of currents in a unit of the axis of the solenoid.
437.] To find the effect of the whole circuit whose element is Oj
on the extremity of the solenoid, we must change the sign of the
above and put a^ = y^; therefore the effect is
_ CAaa^ r Vygyg
2x J Ty% '
an integral of the species considered in § 432 whose value is easily
assigned in particular cases.
438.] Suppose the conductor to le straight, and indefinitely extended
in both directions.
Let ho be the vector perpendicular to it from the extremity of
the canal, and let the conductor be || 77, where Td = Tri = 1 .
Therefore yg = h6+yr} (where y is a scalar),
TyoVo = A/J'rie,
and the integral in § 436 is
hr-qej
J —CD
-.=w.
-00 {h-'+y^f h
The whole effect is therefore
xh
and is thus perpendicular to the plane passing through the conductor
and the extremity of the canal, and varies inversely as the distance of
the latter from the conductor.
254
QUATERNIONS.
[439-
This is exactly the observed effect of an indefinite straight current
on a magnetic pole, or particle of free ina;gnetism.
439.] Suppose the conductor to be circular, and the pole nearly in its
axis.
Let UPD be the conductor, A£ its axis, and C the pole ; £C
perpendicular to A£, and small in comparison with AE = h the
radius of the circle.
Let AJB be Oji,
where
BC=hk, AP = h{jx + i
wJ 'sm-" '•sm.-'
Then
cc/j/^
CP = y =. aii-\-bk—h{jx-\-ky).
• [Fyy
And the effect on C<x -^fy ,
''6' {{h—by)i+a^coJ + aiyk}
{a\-^b'^-^h'^-1bhy)i
where the integral extends to the whole circuit.
440.] Suppose in particular C to be one pole of a small magnet
or solenoid CC whose length is 2 1, and whose middle point is at Q
and distant a from the centre of the conductor.
Let LGGB = A. Then evidently
a^=- a + l cos A,
i = ^ sin A.
Also the effect on C becomes, i£ al + b^+h'^ = A',
^J ^{{h-by)i^a^x3^a^h\ (l + _/ + _ —^ + ...}
15 hHH
Sbh
A^
ia-JjJe
"•" "a^ '^Y a*
+
...),
44I-] PHYSICAL APPLICATIONS. 255
since for the whole circuit
/ey +1 = 0,
f&xy'^ = 0.
If we suppose the centre of the magnet fixed, the vector axis of
the couple produced by the action of the current on C is
IV. {i cos A + ^ sin ^)j-M-
If A, &c. be now developed in powers of I, this at once becomes
■77^^^ sin A .C 6 a^ cos A ISa^^^cos^A 3P
(a2 + /J2)f -^1 ~ a'^ + A^ + (a^ + A^)" '^T^
SlHin^A 15 A^Psin^A _ {a + lcosA)lcosA y 5 a^ cos A n |
" a^ + A^ +T (a2+F)2~ ^^^^ y ~ a'-^-A'^ >V
Putting —I for I and changing the sign of the whole to get that
for pole C , we have for the vector axis of the complete couple
4TrA2;sinA.f ^ ^2(4a2_F)(4-5 sin^A) )
which is almost exactly proportional to sin A if la = A and I be
small.
On this depends a modification of the tangent galvanometer.
(Bravais, Ann. de CAimie, xxxviii. 309.)
441.] As before, the effect of an indefinite solenoid on a^ is
GAaa^^ Va-^y
Now suppose a^ to be an element of a small plane circuit, 8 the
vector of the centre of inertia of its area, the pole of the solenoid
being origin.
Let y = 8 + jO, then a^ = p.
The whole effect is therefore
_ CAaai f r{b + p)p'
2« 7 T{5+pf
_ CAA^aa^ / 38^>.
where A^ and e^ are, for the new circuit, what A and e were for the
former.
Let the new circuit also belong to an indefinite solenoid, and
let 6o be the vector joining the poles of the two solenoids. Then
the mutual effect is
256 QUATERNIONS. [442.
2xx^ J ^m "^ n' )
_ CAA-^aa^ \ Ub^
- 2wx, {n^^°^{n^'
which is exactly the mutual effect of two magnetic poles. Two finite
solenoids, therefore, act on each other exactly as two magnets, and the
pole of an indefinite solenoid acts as a particle of free magnetism,
442.] The mutual attraction of two indefinitely small plane closed
circuits, whose normals are e and e^, may evidently be deduced by
TTh
twice diflFerentiating the expression -f=j-^ for the mutual action of
the poles of two indefinite solenoids, making db in one differentiation
II f and in the other || e^.
But it may also be calculated directly by a process which will
give us in addition the couple impressed on one of the circuits by
the other," supposing for simplicity the first to be circular.
Let A and B be the centres of inertia of the areas of A and B,
« and e^ vectors normal to their planes, o- any vector radius of B,
AB = p.
Then whole effect on </, by §§ 432, 435,
'Tifi + crf
5 r^'^ I
r/3
'V+ Tifi + .f V
^7^i^'^Hl + l^)+ Tl3^ i^ + ^^J
But between proper limits,
frir'rtSdu =-A:ir,r]re€^,
for generally fn'n 86<r = -k{ Fr,crSda- +7.7,7. QfT<T<j').
Hence, after a reduction or two, we find that the whole force
exerted by A on the centre of inertia of the area of B
443-J PHYSICAL APPLICATIONS. 257
This, as already observe^ may be at once found by twice differ-
entiating m;52' ^^ ^^® same way the vector moment, due to A,
about the centre of inertia of £,
These expressions for the whole force of one small magnet on the
centre of inertia of another, and the couple about the latter, seem
to be the simplest that can be given. It is easy to deduce from
them the ordinary forms. For instance, the whole resultant couple
on the second magnet
oc ^
T0^
may easily be shewn to coincide with that given by Ellis {Camh.
Math. Journal, iv. 95), though it seems to lose in simplicity and
capability of interpretation by such modifications.
443.] The above formulae shew that the whole force exerted by
one small magnet M, on the centre of inertia of another m, consists
of four terms which are, in order,
1st. In the line joining the magnets, and proportional to the cosine
of their mutual inclination.
2nd. In the same line, and proportional to five times the product of
the cosines of their respective inclinations to this line.
3rd and 4th. Parallel to { ]■ and proportional to the cosine of the
M ....
inclination o/" { ^ to the joining line.
All these forces are, in addition, inversely as the fourth power of
the distance between the magnets.
For the couples about the centre of inertia of m we have
1st. A couple whose axis is perpendicular to each magnet, and which
is as the sine of their mutual inclination.
.2nd. A couple whose axis is perpendicular to m and to the line
joining the magnets, and whose moment is as three times the product of
the sine of the inclination qfm, and the cosine of the inclination o^M,
to the joining line.
In addition these couples vary inversely as the third power of the
distance between the magnets.
258 QUATEENIONS. [444.
[These results afford a good example of what has been called the
internal nature of the methods of quaternions, reducing, as they do
at once, the forces and couples to others independent of any lines of
reference, other than those necessarily belonging to the system
under consideration. To shew their ready applicability, let us take
a Theorem due to Gauss.]
444.] If two small magnets he at right angles to each other, the
moment of rotation of the first is anproximately twice as great when the
axis of the second passes through the centre of the first, as when the
axis of the first passes through the centre of the second.
In the first case e |{ y3 J.ej^ ;
C 2 C"
therefore moment = ^T(efi-3€€i) = ^yeej.
In the second eil|/3±e;
C
therefore moment = -=j-^Tee-^. Hence the theorem.
445.] Again, we may easily reproduce the results of § 442, if for
the two small circuits we suppose two small mag^nets perpendicular
to their planes to be substituted. (3 is then the vector joining the
middle points of these magnets, and by changing the tensors we
may take 2e and 2ej^ as the vector lengths of the magnets.
Hence evidently the mutual effect
which is easily reducible to
as before, if smaller terms be omitted.
If we operate with V. e^ on the two first terms of the unreduced
expression, and take the difference between this result and the same
with the sign of e^ changed, we have the whole vector axis of the
couple on the magnet 2ei, which is therefore, as- before, seen to be
proportional to
446.] We might apply the foregoing formulae with great ease
to other cases treated by Ampere, De Montferrand, &c. — or to two
finite circular conductors as in Weber's Dynamometer — but in
general the only difficulty is in the integration, which even in some
of the simplest cases involves elliptic functions, &c., &c. {Quarterly
Math. Journal, 1860.)
448-] PHYSICAL APPLICATIONS. 259
447.] Let F{y) be the potential of any system upon a unit
particle at the extremity of y.
F{y) = c (1)
is the equation of a level surface.
Let the differential of (1) be
Svdy=0, (2)
then v is a vector normal to (1), and is therefore the direction of the
force.
But, passing to a proximate level surface, we have Svby = bC. '
Make by=xv, then —a;Tv^ = bC,
Hence v expresses the force in magnitude also. (§ 368.)
Now by § 435 we have for the vector force exerted by a small
plane closed circuit on a particle of free magnetism the expression
A , ZySye\
omitting the factors depending on the strength of the current and
the strength of magnetism of the particle.
Hence the potential, by (2) and (1),
oc ' -
oc
Ty^
area of circuit projected perpendicular to y
oc spherical opening subtended by circuit.
The constant is omitted in the integration, as the potential must
evidently vanish for infinite values of Ty.
By means of Ampere's idea of breaking up a finite circuit into
an indefinite number of indefinitely small ones, it is evident that
the above result may be at once ex-
tended to the case of such a finite closed
circuit.
448.] Quaternions give a simple me-
thod of deducing the well-known pro-
perty of the Magnetic Curves.
Let A, A be two equal magnetic
poles, whose vector distance, 2 a, is bi-
sected in 0, QQ' an indefinitely small
magnet whose length is Ip , where p-= OP^ Then evidently, taking
moments,
S 2
260 QUATEENIONS. [449.
r{p+a)p' _ r{p-a)p'
T{p + af - ± T{p-aY'
where the upper or lower sign is to be taken according as the poles
are like or unlike.
Operate by S. Vap,
Sap{p + af—Sa{p + a)Sp'{p-ira) ^ , -.t, i
^^^ y(p^„)S = ± {s^°ie With -a],
r
or S.af {-^-^V{p-\-a)-= + {same with —a\,
^p + a^
i.e. SadU(p + a) = + SadU{p — a),
Sa { U{p + a) + U{p—a)} = const.,
or cos Z OAP ± cos / OA'P = const.,
the property referred to.
If the poles be unequal, one of the terms to the left must be
multiplied by the ratio of their strengths.
4(49.] K the vector of any point be denoted by
p = ix+Jt/ + iz, (1)
there are many physically interesting and important transformations
depending upon the effects of the quaternion operator
„ ■ d . d , d ,.
^ = ^^+^^ + ^^^ (')
on various functions of p. When the function of p is a scalar, the
effect of V is to give the vector of most rapid increase. Its effect
on a vector function is indicated briefly in § 369.
450.] We commence with one or two simple examples, which
are not only interesting, but very useful in transformations.
7 *
V/) = fiy- +&c.)(«« + &c.) =— 3, (3)
ViTpf = n{TpY-'^VTp = n{TpY-^p; (5)
and, of course, v^-^ = -^^^; (5)i
Tp Tp^ ~ Tp"^
whence, V ^^ =- ^rj =- ^rr. (6)
and, of course, V2y- = — VyY= (6)^
Also, Vp =-3 = TpVUp + VTp.Up = TpVUp-l,
■'■ ^^P = -T^ (7)
453-J PHYSICAL APPLICATIONS. 261
451.] By the help of the above results, of which (6) is especially
useful (though obvious on other grounds), and (4) and (7) very
remarkable, we may easily find the effect of V upon more complex
functions.
Thus, VSap=-V{aic-{-kc.) = -a, (1)
Vrap = — VFpa =—V{pa—Sap) = 3a— a= 2a (2)
Hence
^ ^ap _ 2a ZpVap _ 2ap^ + 3pF'ap _ ap^ — 3pSap . .
T^^Tp^~'T^~ Tf^ " Tp^ ^'
Hence
„ Vap p^ Sahp — ZSapSphp Sahp ZSapSpbp » '^"P / \
'^•8PV^= jp =_______ = _6_.(4)
This is a very useful transformation in various physical applica-
tions. By (6) it can be put in the sometimes more convenient form
S.hpV^=hS.aVy~ (5)
And it is worthy of remark that, as may easily be seen, —S may be
put for V in the left-hand member of the equation.
452.] We have also
'f7r.0py=V{^Syp-pSPy + ySpp] =-yfi + 3S^y-l3y ^SjSy. (1)
Hence, if <j) be any linear and vector function of the form
(j)p = a + ^F.fipy + mp, (2)
i.e. a self-conjugate function with a constant vector added, then
V(f>p = 2S^y—3m = scalar (3)
Hence, an integral of
Vo- = scalar constant, is <t = (l>p (4)
If the constant value of Vo- contain a vector part, there will be
terms of the form Fep in the expression for a; which will then ex-
press a- distortion accompanied by rotation. (§371.)
Also, a solution of V^" = « (where q and a are quaternions) is
q = SCp+Ffp + (Pp.
It may be remarked also, as of considerable importance in phy-
sical applications, that, by (1) and (2) of § 451,
V{S+ir)ap = 0,
but we cannot here enter into details on this point.
453.] It would be easy to give many more of these transforma-
tions, which really present no difiiculty ; but it is sufficient to shew
262 QUATERNIONS. [454.
the, ready applicability to physical questions of one or two of those
already obtained ; a property of great importance, as extensions of
mathematical physics are far more valuable than mere analytical or
geometrical theorems.
Thus, if (7 be the vector-displacement of that point of a homo-
geneous elastic solid whose vector is p, we have, j» being the con-
sequent pressure producedj
Vj9-)-W = 0, (1)
whence <S'SpV^<j-= —SbpVp = 8jb, a complete differential (2)
Also, generally, p = kSVa,
and if the solid be incompressible
S^cT= (3)
Thomson has shewn {Caml. and Bub. Math. Journal, ii. p. 62),
that the forces produced by given distributions of matter, electricity,
magnetism, or galvanic currents, can be represented at every point
by displacements of such a solid producible by external forces. It
may be useful to give his analysis, with some additions, in a qua-
ternion form, to shew the insight gained by the simplicity of the
present method.
454. j Thus, if Scrbp = 8 =,- , we may write each equal to
-stpv^^.
This gives (T = —Vyj^,
J-P
the vector-force exerted by one particle of matter or free electricity
on another. This value of o- evidently satisfies (2) and (3).
Again, if S.hpVa = 6 j—g , either is equal to
-8.hpV^ by (4) of §451.
Here a particular case is
Fap
which is the vector-force exerted by an element a of a current upon
a particle of magnetism at p. (§ 436.)
455.] Also, by §451 (3),
Vap _ ap^ — ZpSap
4'58.j PHYSICAL APPLICATIONS. 263
and we see by §§ 435, 436 that this is the vector-force exerted by a
small plane current at the origin (its plane being perpendicular to a)
upon a magnetic particle, or pole of a solenoid, at p. This expres-
sion, being a pure vector, denotes aii elementary rotation caused by
the distortion of the solid, and it is evident that the above value of
(T satisfies the equations (2), (3), and the distortion is therefore pro-
ducible by external forces. Thus the effect of an element of a
current on a magnetic particle is expressed directly by the displace-
ment, while that of a small closed current or magnet is represented
by the vector-axis of the rotation caused by the displacement.
456.] Again, let ^5pVV=8^.
It is evident that a- satisfies (2), and that the right-hand side of the
above equation may be written
Va,:
o.opv
Hence a particular case is
-^.8pv0^
and this satisfies (3) also.
Hence the corresponding displacement is producible by external
forces, and Vo- is the rotation axis of the element at p, and is seen
as before to represent the vector-force exerted on a particle of mag-
netism at p by an element a of a current at the origin.
457.] It is interesting to observe that a particular value of o- in
this case is ^
(T — —\VSaUp—yjr'
as may easily be proved by substitution.
Again, if Sbpa- = — 8 ~^ >
we have evidently o- = V -jfj •
Now, as yj^ is the potential of a. small magnet a, at the origin,
on a particle of free magnetism at p, o- is the resultant magnetic
force, and represents also a possible distortion of the elastic solid
by external forces, since Vo- = V^o- = 0, and thus (2) and (3) are
both satisfied.
458.] We conclude with some examples of quaternion integra-
tion of the kinds specially required for many important physical
problems.
264 QUATERNIONS. ^ [459.
It may perhaps be useful to commence with a different form
of definition of the operator V, as we shall thus, if we desire it,
entirely avoid the use of ordinary Cartesian coordinates. For this
purpose we write
where a is any unit-vector, the meaning of the right-hand opei'ator
(neglecting its sign) being the rate of change of the function to which
it is applied per unit of length in the direction of the unit- vector a.
If a be not a unit-vector we may treat it as a vector-velocity, and
then the right-hand operator means the- m^e of change per unit of
time due to the change of position.
. Let a, /3, y be any rectangular system of unit-vectors, then by a
fundamental quaternion transformation
V = — aSaV — ySiS/SV — ySyV = ad^ + ^d^ ^ ydy ,
which is identical with Hamilton's form so often given above.
(Lectures, § 620.)
459.] This mode of viewing the subject enables us to see at once
that the effect of applying V to any scalar function of the position
of a point is to give its vector of most rapid increase. Hence, when
it is applied to a potential u, we have
Vu = vector-force at p.
It u be a velocity-potential, we obtain -the velocity of the fluid
element at p ; and if w be the temperature of a conducting solid we
obtain the flux of heat. Finally, whatever series of surfaces is repre-
sented by u = C,
the vector Vu is the normal at the point p, and its length is inversely
as the normal distance at that point between two consecutive sur-
faces of the series.
Hence it is evident that
S.dpVu =—du,
or, as it may be written,
—S.dpV= d;
the left-hand member therefore expresses total differentiation in
virtue of any arbitrary, but small, displacement dp.
460.] To interpret the operator V.aV let us apply it to a poten-
tial function u. Then we easily see that u may be taken under
the vector sign, and the expression
F{aV)u = Y.aSJu
denotes the vector, couple due to the force at p about a point whose
relative vector is o.
462.] PHYSICAL APPLICATIONS. 265
Again, if o- be any vector function of p, we have by ordinary
quaternion operations
r(aV).(r = S.arT7(T-\.a£Vc7 — VSa<T.
The meaning of the third term (in which it is of course understood
that V operates on n- alone) is obvious from what precedes. It
remains that we explain the other terms.
461.] These involve the very important quantities (not operators
such as the expressions we have been hitherto considering),
S.V(T and V.V<t,
which form the basis of our investigations. Let us look upon <t as
the displacement, or as the velocity, of a point situated at p, and
consider the group of points situated near to that at p, as the quan-
tities to be interpreted have reference to the deformation of the
group.
462.] Let T be the vector of one of the group relative to that
situated at p. Then after a small interval of time t, the actual
coordinates become p + i^c
and p + r+t{(T—8{TV)a)
by the definition of V in § 458. Hence, if be the linear and vector
function representing the deformation of the group, we have
^r = T—tS{TV)<T.
The farther solution is rendered veiy simple by the fact that we
may assume t to be so small that its square and higher powers
may be neglected.
If <^' be the function conjugate to <^, we have
^'t = T—tVST<T.
Hence <^r = i(<^ + (^')r + i(0 — c^')''
= t--[-s(tv)o-+ v-Sro-]— ^ r.Trv(T.
The first three terms form a self-conjugate linear and vector func-
tion of r, which we may denote for a moment by utt. Hence
(j)T = ■^r—rf'.rVVa;
or, omitting f^ as above.
Hence the deformation may be decomposed into — (1) the pure strain
■ST, (2) the rotation t „
2
Thus the vector-axis of rotation of the group is
266 QUATERNIONS. [463.
If we were content to avail ourselves of the ordinary results of
Cartesian investigations, we might at once have reached this con-
clusion by noticing that
v% dzJ •'\dz Ax' \dx df
and remembering as in (§ 362) the formulae of Stokes and Helmholtz.
463.] In the same way, as
SV<T=—— — — — —
dx dy dz^
we recognise the cubical compression of the group of points considered.
It would be easy to give this a more strictly quaternionic form by
employing the definition of § 458. Butj working with quaternions,
we ought to obtain all our results by their help alone ; so that we
proceed to prove the above result by finding the volume of the
ellipsoid into which an originally spherical group of points has been
distorted in time t.
For this purpose, we refer again to the equation of deformation
and form the cubic in ^ according to Hamilton's exquisite process.
We easily obtaiuj remembering that <^ is to be neglected*,
(i = ^^-{% — tSV(i)<^^ + {^ — nSV<j)^—{\—tSV<T),
or = (^-1)2(^—1 + i!5Vcr).
The roots of this equation are the ratios of the diameters of the
ellipsoid whose directions are unchanged to that of the sphere.
Hence the volume is increased by the factor
1— i!5Vo-,
from which the truth of the preceding statement is manifest.
* Thus, in Hamilton's notation, X, ;*, v being any three non-coplanar vectors, and
m, m, , «i2 ^^ coefficients of the cubic,
— ttSXnv = S-ip'f^^'iup'v
=8.(\-ty8K<T)(ii-tVSiia){v-tVSr<r)
^S.(\-tVSKa)(yiiv-tVii^8va + tVy'78ii<T)
=8.\iiv-t[S.iivV8\<T + 8.v\'78n<r + 8.\ii'VSva'i
= S.\iiv-t8. l\8.iivV + ii8.v\V + v8.\iiV'\ a
^8.\iiv-tS.\iJiv8Va.
'miS.\iiv=S.\(t>'ii<l)'v + 8.fi^'v<l>'k + 8^vip'\(p'fi
=8.K (ynv-tVnV8vtr+ tVvV8na) + &c.
=8.Kiiv-t8.KnV8va-t8.v\V8n<i-¥ko.
= 38.\nv-2t8Va8.\iiv.
—m^S .\nv = 8 .\ii<p'v + 8. iiv^'\ + 8.v\(j> n
=8.\iJi.v—t8.K/iV8va + &c.
= S8.\iiv—t8V<r8.\iiv
465.] PHYSICAL APPLICATIONS. 267
464.] As the process in. last section depends essentially on the
use of a non-conjugate vector function, with which the reader is less
likely to be acquainted than with the more usually employed forms,
I add another investigation.
Let ■BT = ^T = T—tS{TV)(r.
Then t = (f - V = t:j + tS (in- V) a.
Hence since if, before distortion, the group formed a sphere of radius
1, we have Tt = 1,
the equation of the ellipsoid is
T{'!!T + tS(:!!TV)<T)= 1,
or ■!!r^ + 2iS-nTVS^a- = — 1.
This may be written
S.wx^ = S.w {nr + i VSi!7(T + tS (in- V) <t) — — I,
where x is now self- conjugate.
Hamilton has shewn that the reciprocal of the product of the
squares of the semiaxes is
— 'S'-XWX'^.
whatever rectangular system of unit-vectors is denoted by i, j, h.
Substituting the value of x, we have
—8.{i^tVSi(T^t8{iV)a) (y + &c.) (/^-|-&c.)
= —S.{i■^r tVSia + tS («V) a){i+2 tiSVa— iS{iV)a- tVSicr)
^ l+2tSVa.
The ratio of volumes of the ellipsoid and sphere is therefore, as
before, 1
, = 1 - fSVcr.
VI + 2tSV(T
465.] In what follows we have constantly to deal with integrals
extended over a closed surface, compared with others taken through
the space enclosed by such a surface ; or with integrals over a
limited surface, compared with others taken round its bounding
curve. The notation employed is as foUows. If Q, per unit of
length, of surface, or of volume, at the point p, Q being any qua-
ternion, be the quantity to be summed, these sums will be denoted
by f/qds and Jf/qds,
when comparing integrals over a closed surface with others through
the enclosed space ; and by
f/qds and /QTdp,
when comparing integrals over an unclosed surface with others round
its boundary. No ambiguity is likely to arise from the double use of
268 QUATERNIONS. [466.
for its meaning in any case will be obvious from the integral with
which it is compared.
466.J We have just shewn thatj if a- be the vector displacement
of a point originally situated at
p = ix+jy + kz,
then S.Va-
expresses the increase of density of aggregation of the points of the
system caused by the displacement.
467.] Suppose, now, space to be uniformly filled with points, and
a closed surface S to be drawn, through which the points can freely
move when displaced.
Then it is clear that the increase of number of points within the
space 2, caused by a displacement, may be obtained by either of two
processes — by taking account of the increase of density at all points
within 2, or by estimating the excess of those which pass inwards
through the surface over those which pass outwards. These are
the principles usually employed (for a mere element of volume) in
forming the so-called ' Equation of Continuity.'
Let V be the normal to 2 at the point p, drawn outwards, then
we have at once (by equating the two different expressions of the
same quantity above explained) the equation
///S.Vads =//S.<rUvds,
which is our fundamental equation so long as we deal with triple
integrals.
468.] As a first and very simple example of its use, suppose o-
to represent the vector force exerted upon a unit particle at p (of
ordinary matter, electricity, or magnetism) by any distribution of
attracting matter, electricityj or magnetism partly outsidcj partly
inside 2. Then, if P be the potential at p,
<r = VP,
and if r be the density of the attracting matter, &c., at p,
V(T=V^P = 4irr
by Poisson's extension of Laplace's equation.
Substituting in the fundamental equation, we have
4:i:///rds= 4:-nM=//S.VPUvds,
where M denotes the whole quantity of matter, &c., inside 2. This
is a well-known theorem.
469.] Let P and Pj be any scalar functions of p, we can of course
find the distribution of matter, &c., requisite to make either of them
47I-J PHYSICAL APPLICATIONS. 269
the potential at p ; for, if fhe necessary densities be r and i\ re-
spectively, we have as before
Now V (P VPi) = VP VPj + P V^Pi ,
Hence, if in the above formula we put
we obtain
J/fS.VPVPJs = -///PV^P,ds+//PS.VP,Uvds,
= -///P^^'Pds +//P,S.VP Uvds,
which are the common forms of Greenes Theorem. Sir W. Thomson's
extension of it follows at once from the same proof.
470.] If Pj be a many- valued function, but VPj single- valued,
and if 2 be a multiply-connected* space, the above expressions
require a modification which was first shewn to be necessary by
Helmholtz, and first supplied by Thomson. For simplicity, suppose
2 to be doubly-connected (as a ring or endless rod, whether knotted
or not). Then if it be cut through by a surface s, it will become
simply-connected, but the surface-integrals have to be increased by
terms depending upon the portions thus added to the whole surface.
In the first form of Greenes Theorem, just given, the only term
altered is the last : and it is obvious that if jo^ be the increase of P^
after a complete circuit of the ring, the portion to be added to the
right-hand side of the equation is
Pi/fS.VPUvds,
taken over the cutting surface only. Similar modifications are
easily seen to be produced by each additional complexity in the
space 2.
471.] The immediate consequences of Green's theorem are well
' known, so that I take only one instance.
Let P and P^ be the potentials of one and the same distribution
of matter, and let none of it be within 2. Then we have
///{vpyds =f/ps.vpuvds,
so that if VP is zero all over the surface of 2, it is zero all through
the interior, i.e., the potential is constant inside 2. If P be the
velocity-potential in the irrotational motion of an incompressible
fluid, this equation shews that there can be no such motion of the
* Called by Helmholtz, after Eiemann, mehrfach zusammenhdngend. In translating
Helmholtz'a paper {Phil. Mag. 1867) I used the above as an English equivalent. Sir
■W.Thomson in his great paper on Yortex Motion {Trans. B. S-.E. 1868) uses the ex-
pression "multiply-continuous."
270 QUATERNIONS. [472.
fluid unlesB there is a normal motion at some part of the bounding
surface, so long at least as 2 is simply-connected.
Again, if 2 is an equipotential surface,
f/f(ypfd, = Pf/s.vPUvds = Pf//v^Pds
by the fundamental theorem. But there is by hypothesis no matter
inside 2, so this shews that the potential is constant throughout
the interior. Thus there can be no equipotential surface, not in-
cluding some of the attracting matter, within which the potential
can change. Thus it cannot have a maximum or minimum value
at points unoccupied by matter.
472.] If, in the fundamental theorem, we suppose
a- =Vt,
which imposes the condition that
S.V(T = 0,
i.e., that the <r displacement is effected without condensation, it
becomes //S.VrUvds =///S.V^Tds = 0.
Suppose any closed curve to be traced on the surface 2, dividing
it into two parts. This equation shews that the surface-integral is
the same for both parts, the difference of sign being due to the fact
that the normal is drawn in opposite directions on the two parts.
Hence we see that, with the above limitation of the value of a, the
double integral is the same for all surfaces bounded by a given
closed curve. It must therefore be expressible by a single integral
taken round the cui-ve. The value of this integral will presently
be determined.
473.] The theorem of § 467 may be written
///V^Pds =//S.UvVPds =//S{UvV)Pds.
From this we conclude at once that if
^ = iP+JP^ + kP^,
(which may, of course, represent any vector whatever) we have
///V^ad,=//S{UvV)<Tds,
or, if V^o- = T,
///rds=//S{U,V-')rds.
This gives us the means of representing, by a surface-integral, a
vector-integral taken through a definite space. We have already
seen how to do the same for a scalar-integral — so that we can now
express in this way, subject, however, to an ambiguity presently
to be mentioned, the general integral
476.] PHYSICAL APPLICATIONS. 271
where q is any quaternion \^atever. It is evident that it is only
in certain classes of cases that we can exnect a perfectly definite
expression of such a volume-integral in terms of a surface-integral.
474.] In the above formula for a vector-integral there may
present itself an ambiguity introduced by the inverse operation
to which we must devote a few words. The assumption
is tantamount to saying that, as the constituents of a- are the
potentials of certain distributions of matter, &c., those of t are the
corresponding densities each multiplied by 4 tt.
If, therefore, r be given throughout the space enclosed by S,
o- is given by this equation so far only as it depends upon the
distribution within S, and must be completed by an arbitrary vector
depending on three potentials of mutually independent distributions
exterior to 2.
But, if o- be given, t is perfectly definite ; and as
Vo- = V-^Tj
the value of V""^ is also completely defined. These remarks must
be carefully attended to in using the theorem above : since they
involve as particular cases of their application many curious theorems
in Fluid Motion^ &c.
475.] As a particular case, the equation
rV(r=
of course gives V a- := u, a scalar.
Now, if V be the potential of a distribution whose density is u, we
have V'^v = 4Tr?<.
We know that this equation gives one, and but one, definite value
for V, so that there is no ambiguity in
V = 4tV~^?<,
and therefore o- = — - V« is also determinate.
47r
476.] This shews the nature of the arbitrary term which must
be introduced into the solution of the equation
rVcr=r.
To solve this equation is (§ 462) to find the displacement of any
one of a group of points when the consequent rotation is given.
Here -SVr = -S. V FVo- = 5 W = ;
so that, omitting the arbitrary term (§ 475), we have
W=Vr,
and each constituent of o- isj as above, determinate.
272 QUATERNIONS. [477.
Thomson * has put the solution in a form which may be written
if we understand by y*( ) dp integrating the term in da; as if y
and z were constants, &c. Bearing this in mind, we have as
verification,
rv<r = i2ri[vTi+fr^dp^
= i{3T+/dpSVr}=T.
477.] We now come to relations between the results of integra-
tion extended over a non-closed surface and round its boundary.
Let IT be any vector function of the position of a point. The
line-integral whose value we seek as a fundamental theorem is
yS.adr,
where t is the vector of any point in a small closed curve, drawn
from, a point within it, and in its plane.
Let o-Q be the value of a- at the origin of t, then
a- = <rf,-S(TV)crQ,
so that /S.o-dr =z/S.(a-o-SiTV)<io)dr.
But fdr = 0,
because the curve is closed ; and (Tait on Mectro-Di/namics, § 1 3,
Quarterly Math. Journal, Jan. 1860) we have generally
fS.TVS.Oadr = \S.V{TScr^T-<Tjr.TdT).
Here the integrated part vanishes for a closed circuit, and
\fT.TdT = dsUv,
where ds is the area of the small closed curve, and Uv is a unit-
vector perpendicular to its plane. Hence
fS.cTf^dT = S.V(TgUv.ds.
Now, any finite portion of a surface may be broken up into small
elements such as we have just treated, and the sign only of the
integral along each portion of a bounding curve is changed when
we go round it in the opposite direction. Hence, just as Ampere
did with electric currents, substituting for a finite closed circuit
a network of an infinite number of infinitely small ones, in each
contiguous pair of which the common boundary is described by
equal currents in opposite directions, we have for a finite unclosed
surface /S.adp = jyS.Vcrllv.ds.
There is no diflSculty in extending this result to cases in which the
* Electrostatics and Magnetism, § 521, or Phil. Trans., 1852.
478. J PHYSICAL APPLICATIONS. 273
bounding curve consists offletached ovals, or possesses multiple
points. This theorem seems to have beeu first given by Stokes
(Smith's Prize Esoam. 1854), in the form
=//K'(|-f)-(£-£)+«(l-S))-
It solves the problem suggested by the result of § 472 above.
478.] If a- represent the vector force acting on a particle of
matter at p, —S.adp represents the work done while the particle is
displaced along dp, so that the single integral
/S.adp
of last section, taken with a negative sign, represents the work
done during a complete cycle. When this integral vanishes it is
evident that, if the path be divided into any two parts, the work
spent during the particle's motiou through one part is equal to that
gained in the other. Hence the system of forces must be con-
servative, i. e., must do the same amount of work for all paths
having the same extremities.
But the equivalent double integral must also vanish. Hence a
conservative system is such that
//dsS.V<TUv = 0,
whatever be the form of the finite portion of surface of which ds is
an element. Hence, as Vo- has a fixed value at each point of space,
while Uv may be altered at will, we must have
rvo- = 0,
or Vo- = scalar.
If we call X, T, Z the component forces parallel to rectangular
axes, this extremely simple equation is equivalent to the well-known
conditions
dX_dY_Q ^_^^o ^_^=o
Hy dx " ' dz dy ^ ' dx dz
Returning to the quaternion form, as far less complex^ we see that
Vo- = scalar = 4Trr, suppose,
implies that o- = VP,
where P is a scalar such that
V2P= ^-nr;
that is, P is the potential of a distribution of matter, magnetism, or
statical electricity, of volume-density /.
274 QUATERNIONS. [479.
Hence, for a non-closed path, under conservative forces
-fS.a-dp = -fS.VPdp
= -/S{dpV)P
= /da,P=/dP
= Pi-Po,
depending solely on the values of P at the extremities of the path.
479.] A vector theorem, which is of great use, and which cor-
responds to the Scalar theorem of § 473, may easily be obtained.
Thus, with the notation already employed,
/V.adr =/r{<T,-S(TV)<r,)dr,
Now r{F.vr.TdT)(ra=-S{TS7)r.<TgdT-S{dTV)FT(ro,
and d{S(,TV)r<ToT) = S{TV)r.<T^dT-\-S{dTV)ro^T.
Subtracting, and omitting the term which is the same at both
limits, we have fV, adr = — ¥.(¥. UvV) cr^ ds.
Extended as above to any closed curve, this takes at once the form
/r.(Tdp= -//ds r. ( r. Uvv) <t.
Of course, in many cases of the attempted representation of a
quaternion surface-integral by another taken round its bounding
curve, we are met by ambiguities as in the case of the space-
integral, § 474 : but their origin, both analytically and physically,
is in general obvious.
480.] If P be any scalar function of p, we have (by the process
of § 477, above)
/Pdr =/{P,-S{rV)P,)dT
= -/S.TVPo.dT.
But r.W.rdT = drS.TV—TS.dTV,
and dirSrV) = drS.TV + TS.drV.
These give
/Pdr = -^ {TSTV-F.FTdTV)Po = dsF.UvVPg,
Hence, for a closed curve of any form, we have
/Pdp=//dsr.Uvvp,
from which the theorems of §§ 477, 479 may easily be deduced.
481.] Commencing afresh with the fundamental integral
///SV<rds=//S.aUvds,
put a = UjS,
and we have ///S^Vuds =//uS.^U'vds;
483-] PHYSICAL APPLICATIONS. 275
from which at once ///Vuas = f/uUvds, (1)
or //fVTd,=/fUv.Tds (2)
Putting WjT for r, and taking the scalar, we have
f/f{SrVUi + u.^SVT)ds = f/n^Sr Uvds,
whence ///(S{rV)(r + (T8.VT)ds = //(rSrUvds (3)
483.] As one example of the important results derived from these
simple formulae, take the following, viz. : —
ffr.{Y<j'Uv)Tds = /f<TSTUvds-//UvS<TTds,
where by (3) and (1) we see that the right-hand member may be
written = //f{8{rV)(7 + <rSVT-V S(TT)ds
= -fffr.nv<j)Tds (4)
This, and similar formulae^ are easily applied to find the potential
and vector-force due to various distributions of magnetism. To
shew how this is introduced, we briefly sketch the mode of expressing
the potential of a distribution.
483.] Let or be the vector expressing the direction and intensity
of magnetisation, per unit of volume, at the element ds. Then if
the magnet be placed in a field of magnetic force whose potential
is u, we have for its potential energy
E = -ff/ScrVuds
= ///uSV(rds-//tiS(rUvds.
This shews at once that the magnetism may be resolved into a
volume-density <S(V<7), and a surface-density —ScrUv. Hence, for a
solenoidal distribution, S.'V(r = 0.
What Thomson has called a lamellar distribution (PMl. Trans.
1852), obviously requires that
Sadp
be integrable without a factor ; i. e., that
FVa- = 0.
A complex lamellar distribution requires that the same expression
be integrable by the aid of a factor. If this be u, we have at once
FV[ua) = 0,
or S.<tV(t=0.
With these preliminaries we see at once that (4) may be written
//F.{rcrUv)Tds=-///r.TFV(Tdi-///r.<TVTds+///Sav.Tds.
Now, if T = V(-),
where r is the distance between any external point and the element
276 QUATERNIONS. [484-
ds, the last term on the right is the vector-force exerted by the
magnet on a unit-pole placed at the point. The second term on
the right vanishes by Laplace's equation, and the first vanishes as
above if. the distribution of magnetism be lamellar, thus giving
Thomson's result in the form of a surface integral.
484.] An application may be made of similar transformations to
Ampere's Directrice de V action electrodynamique, which, § 432 above,
is the vector-integral C^pdp
P-
where dp is an element of a closed circuit, and the integration
extends round the circuit. This may be written
-/
r.{dpv)l,
so that its value as a surface integral is
jjs {UvV)V-ds -JJuvV^ i ds.
Of this the last term vanishes, unless the origin is in, or infinitely
near to, the surface over which the double integration extends.
The value of the first term is seen (by what precedes) to be the
vector-force due to uniform normal magnetisation of the same
surface.
2
485.] Also, since VUp = — -^ >
we obtain at once
whence, by difierentiation, or by putting p + a for p, and expanding
in ascending powers of Ta (both of which tacitly assume that the
origin is external to the space integrated through, i.e., that Tp
nowhere vanishes), we have
and this, again, involves
486.] The interpretation of these, and of more complex formulae
of a similar kind, leads to many curious theorems in attraction and
in potentials, Thus, from (1) of § 481, we have
488.] PHYSICAL APPLICATIONS, 277
which.^ves the attraction of a mass of density t in terms of the
potentials of volume distributions and surface distributions. Putting
this becomes
'TJv.<yds
iim-iir-^=fp
Tp JJJ Tp^ ~JJ Tp
By putting cr = p, and taking the scalar, we recover a formula
given above ; and by taking the vector we have
r/fUvUpds = 0.
This may be easily verified from the formula
/Pdp = r//Uv.vPds,
by remembering that VTp = Up.
Again if, in the fundamental integral, we put
(T = tUp,
487.] As another application, let us consider briefly the Stress-
function in an elastic solid.
At any point of a strained body let A. be the vector stress per
unit of area perpendicular to i, n and v the same for planes per-
pendicular to J and k respectively.
Then, by considering an indefinitely small tetrahedron, we have
for the stress per unit of area perpendicular to a unit-vector <a the
expression kSia) + iJ.SJ(o + vSko> =-<j>a>,
so that the stress across any plane is represented by a linear and
vector function of the unit normal to the plane.
But if we consider the equilibrium, as regards rotation, of an
infinitely small parallelepiped whose edges are parallel to i, j, k
respectively, we have (supposing there are no molecular couples)
F{iK+JlJ. + kv) = 0,
or 2 Fi^i = 0,
or r.V^p = 0.
This shews (§173) that in this case (j) is self-conjugate, or, in other
words, involves not nine distinct constants but only six.
488.] Consider next the equilibrium, as regards translation, of
any portion of the solid filling a simply-connected closed space.
Let u be the potential of the external forces. Then the condition
is obviously ff^ ( Vv) ds +fffdiVu = 0,
where v is the normal vector of the element of surface ds. Here
278 QUATEENIONS. [489-
the double integral extends over the whole boundary of the closed
space, and the triple integral throughout the whole interior.
To reduce this to a form to which the method of § 467 is directly
applicable, operate by S.a where a is any constant vector whatever,
and we have /y S .(paUvds + yy/ds SaVu =
by taking advantage of the self-conjugateness of (p. This may be
written ///ds{S.V<t>a + 8.dVv.) = 0,
and, as the limits of integration may be any whatever,
8.V(t>a + S.aVu = (1)
This is the required equation, the indeterminateness of a rendering
it equivalent to tAree scalar conditions.
There are various modes of expressing this without the a. Thus,
if A be used for V when the constituents of <^ are considered, we
may write Vu = -SVA.cjyp.
In integrating this expression through a given space, we must
remark that V and p are merely artificial symbols of construction,
and therefore are not to be looked on as variables in the integral.
489.] As a verification, it may be well to shew that from this
equation we can get the condition of equilibrium, as regards rotation,
of a simply connected portion of the body, which can be written
by inspection as
//r.p<p{Uv)ds+///r.pVuds = 0.
This is easily done as follows : (1) gives
S.V<t>(r + S.crVu = 0,
if, and only if, <r satisfy the condition
S4{V)(T = 0.
Now this condition is satisfied if
cr = Kap
where a is any constant vector. For
S.<p{V)rap=-S.aF<j){V)p
= S.aFV<t)p = 0.
Hence ///'^s {S.Vcj) Fap + S.apVu) = 0,
or f/dsS.apij}Uv+///ds S.apVu = 0.
Multiplying by a, and adding the results obtained by making a in
succession each of three rectangular vectors, we obtain the required
equation.
490.] Suppose a- to be the displacement of a point originally at
p, then the work done by the stress on any simply connected portion
of the solid is obviously
W=//S.(}>{Uv)<Tds,
49I-] PHYSICAL APPLICATIONS. 279
because <j) ( Uv) is the vector Toree overcome per unit of area on the
element ds. This is easily transformed to
W=///S.V(li<7ds.
491. J In this case obviously the strain-function is
X (■nr) = ■ar — /S. (•srV)cr.
Now if the strain be a mere rotation, in which case
S.)(ZlT\T — S.-S7T = 0,
whatever be the vectors ot and t, no work is done by the stress.
Hence the expression for the work done by the stress must vanish
if these conditions are fulfilled.
Again, it is easily seen that when the strain is infinitely small
the work must be a homogeneous function of the second degree of
these critical quantities ; for, if it exist, it is essentially positive.
Hence, even when finite, the work on unit-volume may be ex-
pressed as » = 2.(5.x€X«'- -S""') {S-xrixn'-Sm'),
where e, e', r), rf, which are in general functions of cr, become con-
stant vectors if the stress is indefinitely small. When this is the
case it is easy to see that, whatever be the number of terms under
S, w involves twenty-one separate and independent constants only ;
viz. the coefiicients of the homogeneous products of the second order
of the six values of form
S-XW-XJ—S'STT
iovthe values i, J, ^ of ot or r.
Supposing the strain to be indefinitely small, we have for the
variation of to, the expression
.+ ^{S.x^X^'-S,e'){S.bxr,xri'+S.bxr,'xv)-
Now, by the first equation, we have
SxOT = — *S'(t!rV)8(r.
Hence, writing the result for one of the factors only, the variation
of the whole work done by straining a mass is
bJr= b///wds =/ffbw &
= -^fffd,{8.xy\xr\-Sm) {-S.xe'5.(€V)6<r-f-S.xe^(€'V)8<7}.
Now, if we have at the limits
8(7 = 0,
i.e. if the surface of the mass is altered in a given way, we have
obviously,
fffdsS.'wS{€^)b<T = -///dsS.b(TS{iV)w.
280 QUATERNIONS [492-
Hence
Now any arbitrary change in o- will in general increase the amount
of work done, so that we have
= 2 [5(eV) {x«'('S.X'7X'?'-'Sw')} +'S(e'V) {xeC&XIX'?'-'^'/'?')}].
which is our equation for the determination of cr, as the constants
e, i, t), rj' are dependent solely on the elastic properties of the sub-
stance distorted, and may therefore be considered as known ; while
X essentially involves o-.
492.] Since the algebraic operator
when applied to any function of a;, simply changes x into x-\-U, it
is obvious that if o- be a vector not acted on by
„ . d . d , d
dx •' dy dz
we have ,-s.vy(p) =/(p + ^),
whatever function /"may be. From this it is easy to deduce Taylor's
theorem in one important quaternion form.
If A bear to the constituents of o- the same relation as V bears to
those of p, and if_/and F be any two functions which satisfy the
commutative law in multiplication, this theorem takes the curious
form ,-^^^f{p) F{a) =/(p + A) F{<t) = F{<t + V)f{p) ;
of which a particular case is
,S^f(^,)F{y) =/{x + ±)FQ,) = F{y + ^)/W.
The modifications which the general expression undergoes, when
,/and i''are not commutative, are easily seen.
If one of these be an inverse function, such as, for instance, may
occur in the solution of a linear differential equation, these theorems
of course do not give the arbitrary part of the integral, but they
often materially aid in the determination of the rest.
Other theorems, involving operators such as e*^, e^-'W^ &e., &c.
are easily deduced, and all have numerous applications.
493.] But there are among them results which appear startling
from the excessively free use made of the separation of symbols. Of
these one is quite sufficient to shew their general nature.
Let P be any scalar function of p. It is required to find the
difference between the value of P at p, and its mean value throughout
494-] PHYSICAL APPLICATIONS. 281
a very small sphere, of radius r and volume v, whicli has the ex-
tremity of p as centre.
From what is said above, it is easy to see that we have the fol-
lowing expression for the required result : —
where o- is the vector joining the centre of the sphere with the ele-
ment of volume <?s, and the integration (which relates to o- and &
alone) extends through the whole volume of the sphere. Expanding
the exponential, we may write this expression in the form
higher terms being omitted on account of the smallness of r, the
limit of T<T.
Now, symmetry shews at once that
fff^rd, = 0.
Also, whatever constant vector be denoted by a,
///{Sa^fds = -aV/f{S<rUafds.
Since the integration extends throughout a sphere, it is obvious
that the integral on the right is half of what we may call the
moment of inertia of the volume about a diameter. Hence
{8<TUafd^ = '"^^
///<■
5
If we now write V for a, as the integration does not refer to V,
we have by the foregoing results (neglecting higher powers of r)
l///(.-..v_i)p,,=_ilv^p,
which is the expression given by Clerk-Maxwell*. Although, for
simplicity, P has here been supposed a scalar, it is obvious that in
the result above it may at once be written as a quaternion.
494.J If p be the vector of the element ds, where the surface
density isfp, the potential at o- is
f/dsfpFT{p-<r),
F being the potential function, which may have any form whatever.
By the preceding, § 492, this may be transformed into
ffasfp,^-yFTp;
' London Math. Soc. Proc, vol. iii, no. 3^, 1871.
282 QUATERNIONS. [495-
or, far more conveniently for the integration, into
where A depends on the constituents of a in the same manner as V
depends on those of p.
A still farther simplification may be introduced by using a vector
a-Q, which is finally to be made zero, along with its corresponding
operator Aq, for the above expression then becomes
where p appears in a comparatively manageable form. It is obvious
that, so far, our formulae might be made applicable to any distribu-
tion. We now restrict them to a superficial one.
495.] Integration of this last form can always be easily effected
in the case of a surface of revolution, the origin being a point in
the axis. For the expression, so far as the integration is concerned,
can in that case be exhibited as a single integral
dx<f)Xi''
p
where (f> may be any scalar function, and x depends on the cosine of
the inclination of p to the axis. And
As the interpretation of the general results is a little troublesome,
let us take the case of a spherical shell, the origin being the centre
and the density unity, which, while simple, sufficiently illustrates
the proposed mode of treating the subject.
We easily see that in the above simple case, a being any constant
vector whatever, and a being the radius of the sphere,
/"+" 2 Tra
J —a ■^<*
Now, it appears that we are at liberty to treat A as a has just been
treated. It is necessary, therefore, to find the effects of such opera-
tors as TA, e"^'^, &c., which seem to be novel, upon a scalar function
of To- ; or %, as we may for the present call it.
%F'
Now (rA)2i?'=-A2J = 2?"' + — ,
whence it is easy to guess at a particular form of TA. To be sure
that it is the only one, assume
496.] PHYSICAL APPLICATIONS. 283
where </> and i|f are scalar functions of JC to be found. This gives
= 4>^F" + (<^^' 4- v/'<#) + <i>i') F' + {<pf' + ^^) F.
Comparing, we have
2
(^\/f' + i|'^ = 0.
From the first, ^ = ± Ij
whence the second gives '>/' = + — >
the signs of ^ and \/f being alike. The third is satisfied identically.
That is +yA = ^ + --
~ a® St
Also, an easy induction shews that
±(»)- = (a)"+5(»r
Hence we have at once
by the help of which we easily arrive at the well-known results.
This we leave to the student*.
496.] As an elementary example of the use of V in connection
with the Calculus of Variations, let us consider the expression
A =/QTdp,
where Tdp is an element of a finite are along which the integration
extends, and Q is in general a scalar function of p and constants.
We have bA z=/{bQTdp+QbTdp)
=/{bQTdp- QS. Udpdbp)
= -iqSUdphp-] +/{bQTdp+S.bpdiQUdp)),
where the portion in square brackets refers to the limits only, and
gives the terminal conditions. The remaining portion may easily
be put in the form
S/dp{d{QUdp)-VQ.Tdp).
* Proc. B. S. E., 1871-2.
284 QUATERNIONS. [497-
If the curve is to be determined by the condition that the varia-
tion of A shall vanish, we must have, as 8p may have any direction,
or, with the notation of Chap. IX,
This simple equation shews that
(1) The osculating plane of the sought curve contains the
vector VQ.
(2) The curvature at any point is inversely as Q, and directly as
the component of V Q parallel to the radius of absolute curvature.
497.] As a first application, suppose A to represent the action of
a particle moving freely under a systan of forces which have a
potential*, so that Q := ^o,
and p2 = 2 {P-H),
where P is the potential, H the energy constant.
These give TpVTp = QVQ = -VP,
and qp'= p,
so that the equation above becomes simply
p + VP = 0,
which is obviously true.
498.] If we look to the superior limit only, the first expression
for 6^ becomes in the present case
-{TpSUdptp'] = -Sphp.
If we suppose a variation of the constant H, we get the following
term from the unintegrated part
thH.
Hence we have at once Hamilton's equations of varying action in
the forms y^ _ a
and ^ = t.
The first of these gives, by the help of the condition above,
(VJ)2 = 2 {P-H),
the well-known partial difierential equation of the first order and
second degree.
499.] To shew that, if A be any solution whatever of this equa-
tion, the vector VA represents the velocity in a free path capable of
502.] PHYSICAL APPLICATIONS. 285
being described under the acflon of the given system of forces, we
-j^P = P =-VP=-\V{VAf
= ~S{VA.V)VA.
But ~'VA=-S{fiV)VA.
A comparison shews at once that the equality
VA = p
is consistent with each of these vector equations.
500.] Again, if 5 refer to the constants only,
J a(VJ)2 = S.VA1>VA =-lH
by the differential equation.
But we have also — - = t,
which gives 17^-^) — — 'S'(pV)aJ = 'dH.
These two expressions for 3 jy again agree in giving
VA = p,
and thus shew that the differential coefficients of A with regard to
the two constants of integration must, themselves, be constants.
We thus have the equations of two surfaces whose intersection
determines the path.
501.] Let us suppose next that A represents the time of passagCj
so that the brachistochrone is required. Here we have
the other condition being as in § 497, and we have
which may be reduced to the symmetrical form
p+p-^VP/J = 0.
It is very instructive to compare this equation with that of the free
path as above, § 497.
The application of Hamilton's method may be easily made, as in
the preceding example. (Tait^ Trans. R. S. E., 1865.)
503.] As a particular case, let us suppose gravity to be the only
force, then VP = a,
a constant vector, so that
286 QUATERNIONS. [503-
The form of this equation suggests the assumption
where jo and q are scalars and
Sap = 0.
Substituting, we get
-j)qseo^qt + {-P'>-p'^a^ian^qt) = 0,
which gives joq = T^^ = p^T^a.
Now let jo /3~^o = y ;
this must be a unit-vector perpendicular to a and /3, so that
ir^ = -^— , (cos at— yBm at),
cosqt '• ^ •
whence p = cos qt {cos qt + y sin qt)P~^
(which may be verified at once by multiplication).
Finally, taking the origin so that the constant of integration
may vanish, we have
2/3/8 = t+ — (siQ2g'^— ycos22'^),
2q
which is obviously the equation of a cycloid referred to its vertex.
The tangent at the vertex is parallel to /3j and the axis of symmetry
to a.
503.] In the case of a chain hanging under the action of given
forces Q = Pr,
where P is the potential, r the mass of unit-length.
Here we have also, of course,
/Tdp = I,
the length of the chain being given.
It is easy to see that this leads, by the usual methods, to the
equation -=- {{Pr + ii)p'} —rVP = 0,
where u is a scalar multiplier.
504.J As a simple case, suppose the chain to be uniform. Then
r may be merged in u. Suppose farther that gravity is the only
force, then P = Sap, VP = —a,
and -J- {{Sap+u)p'} +a = 0.
Differentiating, and operating by Sp\ we find
S.p'[p'{8ap'+^)+a'^ = 0;
which shews that u is constant, and may therefore be allowed for
by change of origin.
505.J PHYSICAL APPLICATIONS. 287
The curve lies obviously in% plane parallel to a, and its equation
is {8apY + a^ s^ = const.,
which is a well-known form of the equation of the catenary.
When the quantity Q of § 496 is a vector or a quaternion, we
have simply an equation like that there given for each of the con-
stituents.
505.] Suppose P and the constituents of a- to be functions which
vanish at the bounding surface of a simply-connected space 2, or
such at least that either P or the constituents vanish there, the
others (or other) not becoming infinite.
Then, by § 467,
///d,S.V{Pa) =//dsPSaUv = 0,
if the integrals be taken through and over 2.
Thus ///dsS.(rVP = -///dsPS.V<T.
By the help of this expression- we may easily prove a very re-
markable proposition of Thomson {Cam. and Dub. Maih. Journal^
Jan. 1848, or Reprint of Papers on Electrostatics, § 206.)
To shew that there is one, and lut one, solution of the equation
S.V{e^Vu)= 4ir>-
where r vanishes at anminfinite distance, and e is any real scalar what-
ever, continuous or discontinuous.
Let V be the potential of a distribution of density r, so that
V^v = 4 nr,
and consider the integral
q = —JjJ^s (eVu- -Vv) .
That Q may be a minimum as depending on the value of u (which
is obviously possible since it cannot be negative, and since it may
have any positive value, however large, if only greater than this
minimum) , we must have
= ibQ =-///dsS.(e^Vu—Vv)Vbu
= ///<^s bu S.V {e^Vu-Vv),
by the lemma given above,
=/y/dsbti {S.V {e^Vu)-4:T!r}.
Thus any value of u which satisfies the given equation is such as to
make Q a minimum.
But there is only one value of w which makes Q a minimum ;
for, let Qi be the value of Q when
«j^ = w + (^
is substituted for this value of u, and we have
288 QUATERNIONS. [505-
Qi = —JJJds. (eV (m + <^) - i V w)
The middle term of this expression may, by the proposition at the
beginning of this section, be written
2f//ds<^{SV{e^Vu)-4:T!r},
and therefore vanishes. The last term is essentially positive. Thus
if % anywhere differ from u (except, of course, by a constant quan-
tity) it cannot make Q a, minimum ; and therefore m is a unique
solution
MISCELLANEOUS EXAMPLES.
1. The expression
Fo/3 Fyb + Fay Vh^ + TaS V^y
denotes a vector. What vector ?
2. If two surfaces intersect along a common line of curvature,
they meet at a constant angle.
3. By the help of the quaternion formulae of rotation, translate
into a new form the solution (given in § 234) of the problem of
inscribing in a sphere a closed polygon the directions of whose sides
are given.
4. Express, in terms of the masses, and geocentric vectors of the
sun and moon, the sun's vector disturbing force on the moon, and
expand it to terms of the second order; pointing out the mag-
nitudes and directions of the separate components.
(Hamilton, Lectures, p. 615.)
5. J£ q = r^, shew that
2dq = 2dri = i {dr+Kqdrq-^)Sq-^ = i {dr + q-^drKq)Sq-'-
= (drq + Kqdr)q-''{q + Kq)-^ = {drq + Kqdr){r+Tr)-^
_ dr+Uq-^drUq-^ _ drUq + Uq-^dr _ q-^{U'qdr + drUq-'^)
~ Tq{Uq+Uq-^) ~ q{Uq+Uq-^) " Uq+Uq-^
_ q-^{qdr + Trdrq~'^) _ drUq-{- Uq-^d^ _ drKq-^ +q-'^dr
~ Tq{Uq+Uq-^) ~ Tq{l + Ur) " iTUr
1
MISCELLANEOUS EXAMPLES. 289
2clq =^\clr+ r.Fdrjqlq-^ = j 3r -V.Vdrj q-'' \q-
q q S^ q q S ^
= drq-^ + V. Vq-^ Vcl/r (l + -^ j-i) :
and give geometrical interpretations of these varied expressions for
the same quantity. {Ihid. p. 628.)
6. Shew that the equation of motion of a homogeneous solid of
revolution about a point in its axis, which is not its centre of
gravity, is BYp^-ASlp = Ypy,
where 12 is a constant. {Trans. U. 8. E., 1869.)
7. Integrate the differfential equations :
{a.) % + aq = h,
where a and h are given quaternions, and and -v/f given linear and
vector functions. (Tait, Proe. B.S.E., 1870-1.)
8. Derive (4) of § 92 directly from (3) of § 91.
9. Find the successive values of the continued fraction
where i and j have their quaternion significations, and so has the
values 1, 2, 3, &c. (Hamilton, Lectures, p. 645.)
10. If we have m. = f-A) c,
where c is a given quaternion, find the successive values.
For what values of c does u become constant ? {Ihid. p. 652.)
11. Prove that the moment of hydrostatic pressures on the faces
of any polyhedron is zero, {a.) when the fluid pressure is the same
throughout, {b.) when it is due to any set of forces which have a
potential.
12. What vector is given, in terms of two known vectors, by the
relation p-^ = \ {ar^ + yS'^) ?
Shew that the origin lies on the circle which passes through the
extremities of these three vectors.
tr
290 QUATEENIONS.
13. Tait, Tram, and Proc. R.S.B., 1870-3.
With the notation of §§ 467, 477, prove
(«•) ///S{aV)rds =//rSaUvds.
(6.) I{ S{pV)T = -nT,
(« + 3)///r& = -f/rSp Uvds.
(e.) With the additional restriction V^r = 0,
//S.mi2np+{n+3)p^V).Tds = 0.
(d.) Express the value of the last integral over a non-
closed surface by a line-integral.
(e.) -/Tdp =f/ds8.UvV<T,
if (7 = Udp all round the curve.
{/.) For any portion of surface whose bounding edge lies
wholly on a sphere with the origin as centre
ffds8.{UpUvV).<r = 0,
whatever be the vector o-.
iff.) /rdpV.tr =//ds{UvV^-S{UvV)V)(T,
whatever be o-.
14. Tait, Trans. B. S. U., 1873.
Interpret the equation
d(T = uqdpq~^,
and shew that it leads to the following results
V^cr = qVn q~^,
V.Mj-i = 0,
V^M* = 0.
Hence shew that the only sets of surfaces which, together, cut
space into cubes are planes and their electric images.
1 5. What problem has its conditions stated in the following six
equations, from which ^, rj, ( are to be determined as scalar functions
ot x, y, g, or oi p = is!+jy+kz'>
V^i = 0, V^r, = 0, V^f = 0,
SViVrj = 0, SVriVC= 0, SV^Vi = 0,
„ . d . d , d
where V = »^- + ?^- +/e-=- ■
dx '' dy dz
Shew that they give the farther equations
MISCELLANEOUS EXAMPLES. 291
Shew that (with a change OT origin) the general solution of these
equations may be put in the form
where <j(> is a self-conjugate linear and vector function, and £, rj, (
are to be found respectively from the three values of_/at any point
by relations similar to those in Ex. 24 to Chapter IX. (See Lame,
Journal de MatAematiqties, 1843.)
16. Shew that, if p be a planet's radius vector, the potential P of
masses external to the solar system introduces into the equation of
motion a term of the form S (pV)VP.
Shew that this is a self-conjugate linear and vector function
of p, and that it involves only Jive independent constants.
Supposing the undisturbed motion to be circular, find the chief
effects which this disturbance can produce.
17. In § 405 above, we have the equations
?a(OT + «^OT) = 0, 8a^ =0, d = aiFia, Ta = 1,
where u>^ is neglected. Shew that with the assumptions
bit Uif
qz^i", a = qPq-'^, r = fi", •sr = qrrr-^q-^,
we have /3 = 0, Tj3 = 1, S/3t=0, F0{T + n^T) = O,
provided co*S«a— coj^ = 0. Hence deduce the behaviour of the Fou-
cault pendulum without the x, y, and ^, jj transformations in the
text.
Apply analogous methods to the problems proposed at the end of
§ 401 of the text.
18. Hamilton, Bishop Law's Premmm Examination, 1862.
[a.) If OABP be four points of space, whereof the three first are
given, and not eoUinear ; if also oa = a, ob = /3, op = p ;
and if, in the equation
a a
the characteristic of operation F be replaced by S, the
locus of P is a plane. What plane ?
{i.) In the same general equation, if F be replaced by V, the
locus is an indefinite right line. What line ?
(c.) If F be changed to K, the locus of p is a point. What
point ?
(d.) If F be made = TJ, the locus is an indefinite half-line, or
ray. What raj^ ?
292 QUATERNIONS.
(e.) If F be replaced by T, the locus is a sphere. What sphere ?
{/.) If F be changed to TV, the locus is a cylinder of revo-
lution. What cylinder ?
{g.) If 2?' be made TVU, the locus is a cone of revolution. What
cone ?
[h.) If SU be substituted for F, the locus is one sheet of such a
cone. Of what cone ? and which sheet ?
(«.) If i'' be changed to VU, the locus is a pair of rays. Which
pair?
19. Hamilton, Bishop Law's Premium Examination, 1863.
{a.) The equation Spp' + a^ —
expresses that p and p' are the vectors of two points
p and p', which are conjugate with respect to the sphere
or of which one is on the polar plane of the other.
(b.) Prove by quaternions that if the right line pp', connecting
two such pointSj intersect the sphere, it is cut har-
monically thereby.
(c.) If p' be a given external point, the cone of tangents drawn
from it is represented by the equation,
irppy = a^p-py;
and the orthogonal cone, concentric with the sphere, by
i8ppy+a^p"' = 0.
{d.) Prove and interpret the equation,
T{np-a) = T{p-na\ if Tp = Ta.
{e.) Transform and interpret the equation of the ellipsoid,
y(tp + p/() = K2_t2.
{/.) The equation
{k^-I^Y = {l^ + K^)Spp' + 2SLpKp'
expresses that p and p' are values of conjugate points,
with respect to the same ellipsoid.
(ff.) The equation of the ellipsoid may also be thus written,
S,;p = 1, if {k'^-l^)^v = {i.-kYp+2iSkp+ 2kSip.
{h.) The last equation gives also.
MISCELLANEOUS EXAMPLES. 293
{i.) With the same sigiltfication of v, the differential equations
of the ellipsoid and its reciprocal become
Svdp — 0, Spdu = 0.
{j.) Eliminate p between the four scalar equations,
Sap = a, Spp = b, Syp = c, Sep = e.
20. Hamilton, Bishop Law^s Premium Examination, 1864.
{a.) Let Aj^B-^j A^,^^, ... A^B„ be any given system of posited
right lines, the 2n points being all given; and let
their vector sum,
AB = Aj^B^+A^B^+.-.+A^B^,
be a line which does not vanish. Then a point H, and
a scalar A, can be determined, which shall satisfy the
quaternion equation,
HAj^.A^Bi+... +HA^.A^B^ = h.AB ;
namely by assuming any origin 0, and writing,
Qjj_ jr OA-AA + ■ ■ • + OAn-A„B„
AiB^+...+A„B„
A^B,+ ...
(b.) For any assumed point C, let
Qc = CA^.A^B^ + . . . + CA^.A^B,, ; ,
then this quaternion sum may be transformed as follows,
Qc= Qh + CH.AB = {7i + GH).AB ;
and therefore its tensor is
Tqc = {fi'' + CH^f.lB,
in which AB and CH denote lengths.
(c.) The least value of this tensor TQc is obtained by placing
the point. C at H; if then a quaternion be said to be a
minimum when its tensor is such, we may write
min. Qc = Qj=r= h.AB;
so that this minimum of Qc is a vector.
{d.) The equation
TQc = c = any scalar constant > TQh
expresses that the locus of the variable point C is a
spheric surface, with its centre at the fixed point H,
and with a radius r, or CH, such that '
r.AB = {TQc^-TQH^)i = (c^ - h\ AB^)^ ;
294 QUATERNIONS.
so that H, as being thus the common centre of a series
of concentric spheres, determined by the given system
of right lines, may be said to be the Central Point, or
simply the Centre, of that system.
(e.) The equation
TFQc = Cj = any scalar constant > TQh
represents a right cylinder, of which the radius
divided by AB, and of which the axis of revolution is
the line, VQc = Qh = h.AB;
wherefore this last right line, as being the common
axis of a series of such right cylinders, may be called
the Central Axis of the system.
(/".) The equation
SQc = ^2 = ^°y scalar constant
represents a plane ; and all such planes are parallel to
the Central Plane, of which the equation is
{g.) Prove that the central axis intersects the central plane
perpendicularly, in the central point of the system.
•
(Ji,.) When the n given vectors A^B-i, ... A„B„ are parallel, and
are therefore proportional to n sealars, b^,...6„, the
scalar A and the vector Qh vanish ; and the centre H is
then determined by the equation
bi.HAi+i2SA+--- + h-SA„= 0,
or by the expression,
where is again an arbitrary origin.
21. Hamilton, Bishop Law's Premium Examination, 1860.
{a.) The normal at the end of the variable vector p, to the
surface of revolution of the sixth dimension, which is
represented by the equation
(p2-a2)3 = 27a==(p-a)*, (a)
or by the system of the two equations,
p2_a2=3<2„2^ (p_„)2 = ^3^2^ ^^,^
MISCELLANEOUS EXAMPLES. 295
and the tangent to the meridian at that point, are
respectively parallel to the two vectors,
• v = 2{p-a)-tp,
and T=2{l-2i){p-a) + i^p;
so that they intersect the axis a, in points of which the
vectors are, respectively,
2a 2{l — 2f)a
2-t' -^"^ {2-tf-2
(b.)
If dp be in the same meridian plane as p, then
t{l t){i-f)dp=3Tdi, and s''f* =^~*
up 3
Under the same condition.
(e.)
'^^ = 1(1-^).
dp 3^ '
(d.) The vector of the centre of curvature of the meridian, at
the end of the vector p, is, therefore,
/„<?Dx-^ 3 V 6a — (4 — Op
'' = P-<^dp) =P-2Y^t= 2(1-0 •
{e.) The eiqjressions in Example 38 give
v^ = a?t^{\-tf, T^ = aH%l-f)^4:-f);
9 9 a^t
hence (cr—pY = -t^P', and dp''' = -.dt"^;
the radius of curvature of the meridian is, therefore.
and the length of an element of arc of that curve is
= Tdp=zTa{-^fdt.
{/.) The same expressions give
thus the auxiliary scalar t is confined between the limits
and 4, and we may write t = 2 vers $, where 5 is a
real angle, which varies continuously from to 2Tr ; the
recent expression for the element of arc becomes, there-
fore, ds=3TaJd0*
and gives by integration
s = 6Ta{e-sme),
if the arc s be measured from the point, say F, for which
p = a, and which is common to all the meridians ; and
the total periphery of any one such curve is = 12Tr Ta.
296 QUATERNIONS.
{ff.) The value of o- gives
i{<r^-a^) = 3aH{i-t), 16{Fa<r)^ = -a*f{i-tf ;
if, then, we set aside the axis of revolution o, which is
crossed by all the normals to the surface (a), the surface
of centres of curvature which is touched by all those
normals is represented by the equation,
4 (0-2 -a2)3 + 27 a2(rao-)2 = (b)
{h.) The point F is common to the two surfaces (a) and (b),
and is a singular point on each of them, being a triple
point on (a), and a double point on (b) ; there is also at
it an infinitely sharp cusp on (b), which tends to coincide
with the axis a, but a determined tangent plane to (a),
which is perpendicular to that axis, and to that cusp ;
and the point, say.?", of which the vector =— a, is
another and an exactly similar cusp on (b), but does not
belong to (a).
(j.) Besides the three universally coincident intersections of the
surface (a), with any transversal, drawn through its
triple point F, in any given direction y9, there are
always three other real intersections, of which indeed one
coincides with F if the transversal be perpendicular to
the axis, and for which the following is a general
formula :
p=Ta.[Ua+ {28U{a^)iYU^'].
{j.) The point, say V, of which the vector is p=2a, is a
double point of (a), near which that surface has a cusp,
which coincides nearly with its tangent cone at that
point ; and the semi-angle of this cone is = - •
Auxiliary Equations :
(2Sp{p—a) = aH^{3 + t),
\2Sa{p-a) = aH^ls-t).
f Svp =-aH{l—t){l—2t),
l2Sv{p-a) = aH''{l-f).
( SpT^aH^{l~t){^-t),
l2S(p-a)T= aH^l-t){4:-t).
University Press, Cambridge,
January, 1880.
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NALOPAKHYANAM, OR, THE TALE OF NALA j
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to derived words in Cognate Languages, and a sketch of Sanskrit
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GREEK AND LATIN CLASSICS, &c. (See also pp. 20-23.)
THE AGAMEMNON OF AESCHYLUS.
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HONOURABLE HENRY CAVENDISH, F.R.S.
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HISTORY.
LIFE AND TIMES OF STEIN, OR GERMANY
AND PRUSSIA IN THE NAPOLEONIC AGE,
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the University of Cambridge, with Portraits and Maps. 3 Vols.
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Great,_ Goethe and Stein — the first two found
long since in Carlyle and Lewes biographers
who have undoubtedly driven their German
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year just past Professor Seeley of Cambridge
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presumption of teaching us Germans our own
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liant superiority all that we have ourselves
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THE UNIVERSITY
THE EARLIEST
INJUNCTIONS OF
OF CAMBRIDGE FROM
TIMES TO THE ROYAL
IS3S,
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HISTORY OF THE COLLEGE OF ST JOHN
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Indian Mail.
"Von nicht geringem Werthe dagegen sind
die Beigaben, welche Wright als 'Appendix'
hinter der 'history* folgen iSsst, AufzSh-
lungen namlich der in NepSl (iblichen Musik-
Instrumente, Ackergerathe, Miinzen, Ge-
wichte, Zeittheilung, sodann ein kurzes
Vocabular in Parbatly^ und NewSri, einige
New^rl songs mit Interlinear-Uebersetzung,
eine KOnigsliste, und, last not least, ein
Verzeichniss der von ihm mitgebrachten
Sanskrit-Mss. , welche jetzt in der UniVersi-
tats-Bibliothek in Cambridge deponirt sind."
— A. Weber, LiteraturzeiUing, Jahrgang
1877, Nr. 26.
*' On trouve le portrait et la g^n^alogie
de Sir Jang Bahadur dans I'excellent ouvrage
que vientde publier Mr Daniel Wright
sous le titre de * History of Nepal, translated
from the Parbatiya, etc.'"— M. Garcin db
Tassv in La Langue et la Littirature Hin-
doustanies in 1877. Paris, 1878.
SCHOLAE ACADEMICAE:
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